CYCLE DECOMPOSITIONS OF K , AND Kn - I

by

Mateja Sajna B.Sc.(Hon.), University of Ljubljana, 1992

M.Sc., Simon Fraser University, 1994

Thesis submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the Department of

Mathematics and Statistics

QMateja Sajna 1999 Simon Fraser University

July 1999

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Abstract

When does a complete gaph admit a decomposition into cycles of some fixed

length? Since the existence of such a decomposition requires that the degrees of all

vertices be even, the complete gaph must have an odd number of vertices. However,

this question can be extended to graphs with an even number of vertices in which

every vertex has even degree. A natural way of creating such graphs that are very

'close" to complete graphs is to remove a 1-factor from a complete g a p h with an

even number of vertices. The question now becomes the following: when does K, or K,, - I, whichever is appropriate, admit a decomposition into cycles of a fixed

length m?

There are two obvious necessary conditions, namely, that 3 5 m 5 n and that n n 1 the cycle length m divides the number of edges in either K,,, that is, 9, or

Kn - I, that is, 9. B. Alspach and H. Gavlas [l] have shown that for the case when m and n are

either both odd or both even, the necessary conditions are also sufficient. In this

thesis we extend their results to the case m even, n odd, and m odd, n even. That

is, we give a constructive proof of the following two statements:

a K, - I can be decomposed into cycles of length rn whenever n is even, m is

odd, 3 5 rn < n, and rn divides v; snd

r K,, can be decomposed into cycles of length rn whenever n is odd, m is even,

3 _< rn 5 n, and m divides v.

iii

Acknowledgement

If I have seen further it is by standing on the rhouldm of giants.

Sir Isaac Newton

Thanks to all the giants on whose shoulders I have stood while working on

this thesis, in particular, to B. Alspach and H. Gavlas, whose techniques I have

adopted, adapted, and further developed. Thanks to my supervisor Brian Alspach

for saving this beautiful problem for me and for not giving up on me after so many

unproductive months. Thanks to the Department of Mathematics and Statistics for

financial support and encouragement, in particular, to Luis Goddyn for sharing the

joy of discovery and to Malgonata Dubiel for being the best "boss" (lab instructor)

one can imagine.

I would also like to thank all the people who have made these past few years

so unforgettable. Thanks to dl my dance teachers from SFU's School for the Con-

temporary Arts, fellow dancers, and fellow singers from the SFU choir for sharing

the joy of music and dance, and for teaching me about art and life. Thanks to the

Slovenian community in Vancouver for their warm acceptance. Thanks to Darja, Sanja, Yves, Jocelyn, Lisa, and Shabnam for their friendship. Special thanks to

Michael, who during this time taught me of love and courage, and to Pat who has

held the mirror for so long and helped me get so much closer to all I can be. Finally,

thanks to aU my friends beyond who were there for me whenever I called in need.

Dedication

Mojim stadem, ki niso nikoli podvomili vame.

To my parents, who never lost faith in me.

Foreword: A Personal Note

It was a cool day in February. My seventh year as a graduate student, my fifth

year in the Ph.D. programme. My thesis? Oh well, I had a chunk of work done;

perhaps I'd be able to scrape together a survival thesis with just a little bit more

work. When people asked when I was going to graduate, I'd say: "Hopefully within

a year. But then again, you never know; I said that two years ago, too." At that

point, I'd stopped worrying. I had two months of intense grieving and self-searching

behind me and I was simply glad to be able to think and find pleasure in life again.

That cool Tuesday in February, getting my weekly supply of fresh vegetables in

my favourite produce store, I ran into a friend I hadn't seen for a long time. There

are people I call friends because we share a lot of history and then there are people

I've never spent much time with for some reason although every one of those few

encounters makes me think that we have a lot in common. M. belonged to the

latter category. That February afternoon our conversation included the state of my

research. "I got a lot of work done last summer," I mentioned, "but I've been stuck

for a few months now." For a second I hesitated to reveal my New Age streak. 'Oh

well," I finally sighed, "I guess I'll just keep working to show the Universe I am

serious about this and then trust that it will happen." M.'s reply was a pleasant

surprise. "Have you tried meditating?" he asked. "When I meditate, I often wake

up with fresh insights." I mentioned my unsuccessful attempts at meditating and

then reflected upon my days as a high school math enthusiast going to bed at night

with a problem on my mind and often finding a solution just before drifting off to

sleep. 'But that hasn't happened for a long time," I added with a hint of sadness.

That very night I went to bed with a notepad and a hunch. Before I fell asleep,

the hunch had evolved into an idea. The next day one idea led to another and by

the end of the week the problem I'd been working on for a year was solved.

Two months later my research was completed. (Or so I thought at the time.

Later. just before the defence. I had to do a major revision of a part of the thesis.)

I had proved my two theorems, an amount of work more than enough for a Ph.D.

thesis. This was a magnificent time. As if a channel had opened and the material

kept coming, pouring into my mind. It didn't matter if an idea proved fruitless

because a new one was right there, ripe. It was a hard time, too, my head aching,

bursting with new ideas, ready to explode. Compelled to work, lest I miss something,

I was waking up very early in the morning and not getting enough sleep. Oh, the

bit tersweetness of an obsession!

"So, what happened?" asked Brian. "We worked on that one case, i(loo - I into

Ca5, for SO long and we couldn't get a decomposition. How did it all come together

so quickly then?" 'I started meditating," I replied with a wink but I was only half

joking.

Perhaps it was just a matter of time and of learning to look at the expression for

the number of cycles in a different way, a way that suggests a possible decomposition.

Or was it the change in my attitude, a new determination to do my share of the

work and let the Universe, Higher Power, God, the Tao, the Force, or whatever one

wishes to call It, do Its own? Is it possible that my declaration of this determination

ignited this, as Brian called it, gunfire of results that followed? I don't know. But

I cannot help but feel that there is a mystical dimension to these events, as there is

(in my eyes) to digging into the mysteries of mathematics in general.

This brings me to the question of why I want to share this story. I am well aware

that I might sound pretentious or at least weird. But I've always been inspired by

personal accounts and by philosophical aspects of mathematics, and perhaps my

story will inspire others. It is, after all, a story of failure turned into success. It is a story of years of self-doubt, thinking I wasn't smart enough, focused enough,

weird enough, or perhaps too weird, too sensitive, and who knows what else, to be a

vii

mathematician. It took a long time before I realized that instead of trying to change

my body to fit the clothes I wanted to wear I could alter the clothes to fit my figure.

Moreover, I realized that not only will I be happier doing it my way but also this is

the only way that I can contribute anything unique and original to the world.

And so. dear reader, besides the theorems. this is what I'd like to leave behind.

If I had to sum up the lesson on success I have learnt in the last few months, this

would be it: Remember how you learnt as a small child. Find your natural curiosity

and success will follow effortlessly. I know that I will forget this lesson many times

and so I'm writing this for myself, too, as I will be in the future.

The other lesson I want to make a vow never to forget is the lesson on failure.

There is very real suffering in the state of wanting or having to do math and not

being able to. May all of us who teach mathematics never forget that for many

of our students math is synonimous with "stuckness", frustration, and self-doubt.

May our experience with failure help us become more patient, understanding, and

compassionate teachers.

July 23, 1999

Burnaby, B.C.

And now, to the math of it.

Contents

Approval

Abstract

Acknowledgement

Dedication

Foreword: A Personal Note

Contents

List of Figures

1 Introduction 1.1 Definitions and terminology . . . . . . . . . . . . . . . . . . . . . . . 1.2 The origin of the problem and early results . . . . . . . . . . . . . . . 1.3 Recent results by Alspach and Gavlas . . . . . . . . . . . . . . . . . .

1.3.1 The problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 A brief outline of the constructions by Alspach and Gavlas . .

1.1 A brief discussion of the techniques used in the constructions . . . . . 1.4.1 Peripheral cycles . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Diameter cycles and central cycles . . . . . . . . . . . . . . . . 1.4.3 A brief outline of the constructions . . . . . . . . . . . . . . .

iii

Decomposition of Kn . I into m.cycles. where n is even and m is

odd 17 . . . . . . . . . . . . . . . . . . . 2.1 Inductive step and the main result 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Introduction 17 . . . . . . . . . . . . . . . . . 2.1.2 Some C, -decomposable graphs 19

. . . . . . . . . . . . . . 2.1.3 Proof of the Induction Theorem 2.1.1 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The main result 27

2.2 Cm-decomposition of h;, - I for n even, m odd. and 2m 5 n < 3m . 27

2.3 Cm-decomposition of h:, - I for n even, rn odd, and m 5 n < 2m . . 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Preliminaries 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Central cycles 38

. . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Peripheral cycles 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Conclusion 59

3 Decomposition of Kn into rn.cycles. where n is odd and m is even 61 . . . . . . . . . . . . . . . . . . . 3.1 Inductive step and the main result 61

3.2 C,. decomposition of Kn for n odd. rn even. and 2m < n < 3772 . . . . 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Central cycles 63

. . . . . . . . . . . . . . 3.2.2 Peripheral cycles for the case c 5 9 66

3.2.3 Peripheral cycles for the case c > 9 . . . . . . . . . . . . . . 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Conclusion 90

. . . . 3.3 Cm-decomposition of Kn for n odd. rn even. and m 5 n < 2m 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Preliminaries 91

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The case c < 4 92 d . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Thecasecz? 113

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Conclusion 150

Bibliography 152

List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A graph G and G(2) 2

A zig-zag path with the edge length set { a l . az. .... a,) . . . . . . . . . 5

Ezarnples of central cycles for m and n both even (left). and for m

. . . . . . . . . . . . . . . . . . . and n both odd. m 5 n < 2m (right) 10

... Lemma 2.1.9.- Co in the Cs-decomposition of K9(2)(0. t1.2. . 7 ) . 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 19

Lemma 2.1.4: Co in the C9-decomposition of &(2) ( ( 4 1 , {I, 2,3,4}, 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K2 20

Lemma 2.1.5: Cl and C; in the Cs-decomposition of &(2)({4), { 1 , 2 ) , 0) . 21 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m-paths Po and Po 28

m-paths P and T ( P ) , and a C,- decomposition of G2(P) . . . . . . . . 30 A C,- decomposition of Cm(2) for m = 3 and m 2 5 . . . . . . . . . . 31

Lemma 2.2.9: the zzg-rag (m' - 1)-path Pone for m' 2 5 . . . . . . . . . 33 . . . . . . . . . . . . . . . . . Lemma 2.9.2: the rig-rag mf-path Pop 39

R1 and Rz for Case 1.1: n' O(mod 4)) A odd (subcase N odd shown) . 42

R1 and R2 for Case 1.2: n' a O(mod 4), A even (subcase N odd shown) . 43

. . . . . . . R1 andR2 forCase 2.1.1. n1a2 (mod4) , N even, A odd 44

. . . . . . . R1 and R2 for Case 2.1.2. n' a 2(mod 4)) N even, A even 45

R1 and Rz for Cose 2.2.1: n' i 2(mod 4), N odd, Az odd . . . . . . . 46

. . . . . . . R1 and R2 for Case 2.2.2. n' 6 2(mod 4), N odd) A2 even 48

R1 and Rz for Case I: n' i O(mod 4) (subcase A < 0 shown here) . . . 51

R1 and Rq for Case 2: n' 2 ( m d 4) hubcase A > 0 shorn hem) . . . 52

Lemma 2.9.4: the zig-rug (ml- 1)-path PlVa for mf >_ 5. r' 2 5. m' 5 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (mod4)

Lemma 2.9.4: the rig-zag (m'- 1)-path PIPo for m' 2 5. rf 2 5. m' 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (mod4)

Lemma 2.9.4. the ripzag (m' - 1)-path PI. for r' 5 3. m' 2 5 . . . . .

From a decomposition of K,, to a decomposition of Kn+?rn . . . . . . A C, -decomposition of K2. +l for rn = 0 (mod 4 ) and m r 2 (mod 4 ) . Lemma 5.2.2: the zag-zag m'-path POvo for rn' r 2 (mod 4). m' > 2 . . Lemma 9.2.2. the rig-zag m'-path Po* 0 for m' s 0 (mod 4 ) . . . . . . . Lemma 9.2.9: the rig-zag mf -paths Po. and PtvO for m' E 2 (mod 4).

c > l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lemma 9.2.9. the rig-zag m'-path PoVc for m' = 2 (mod 4) , c = 1 . . . Lemma 3.2.5: the zig-zag m'-paths Po. and P;., for m' a 0 (mod 4),

m'33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lemma 3.4.3. the rig-zag mf-paths Poa and PiD, for m' = 4 . . . . . . Lemma 9.2.9. the central cycle for m' = 4 , c = 1 . . . . . . . . . . . . Lemma 3.9.1: the paths R1 and R2 that generate the central cycles of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the first type

Lemma 9.3.2: the paths R1 and R2 that generate the central cycles of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the second type

. . . . . . . . . . . . . . . . . . . . . . . The crossover &path Q J x . y ) 100 . . . . . . . . . . . . . . . . . . . . . . Lemma 3.3.6. the m'-path PoPo 102

Lemma 9.3.6. the m'-path Po. for m' = 7. nf = 10 . . . . . . . . . . . 103 . . . . . . . . . . . . . . . . . . . . . . Lemma 3.9.7. the m'-path Poqo 105 . . . . . . . . . . . . . . . . . . . . . . Lemma 9.9.8. the m'-path PoPo 107

. . . . . . . . . . . Lemma 3.9.8. the m'-path PoVo for rn' = 13. n' = 14 109 . . . . . . . . . . . . Lemma 9.3.8. the m'-path qp for rn' = 5. n' = 6 109

. . . . . . . . . . . . . . . . . . . . . . Lemma 9.9.9. the mf-path Por 111

. . . . . . . . . 3.20 Lemma 8.9.9. the m'-path Poao for mf = 9 and n' = 10 113

xii

3.21 An ezample of a diameter cycle . . . . . . . . . . . . . . . . . . . . . . 114

3.22 The path R for Case 1 . 1. 1: A - B odd, a and f even . . . . . . . . . 118

3.23 The path R for Case 1.1.'. A - B odd, a even. f odd . . . . . . . . . 120

3.24 The path R for Case 1.2.1. A - B and a odd. even . . . . . . . . . 121 3.25 The path R for Case 1.2.2.- A - B, a! and f odd . . . . . . . . . . . . 122

3.26 The path R for Case 2.1.1. A - B. a and f even . . . . . . . . . . . . 123

3.27 The path R for C u e 2.1.2. A - B and a even. f odd . . . . . . . . . 124

3.28 The path R for Case 2.2.1. A - B even. a odd) f even . . . . . . . . . 125

3.29 The path R for Case 2.2.2. A - B even. a and f odd . . . . . . . . . 126

3.30 The path R for Case 1.1.1: A - B odd. m' n 1 (mod 4 ) ) and a even . . 132

3.31 The path R for Case 1.1.2: A - B odd. rn' I 1 (mod 4). and a odd . . 134

3.32 The path R for Cuse 1.2.1. A - B odd. m' r 3 (mod 4) . a even . . . . 135

3.33 The path R for Case 1.2.2.- A - B odd. rn' n 3 (mod 4)) a odd . . . . . 137

3.34 The path R for Case 2.1.1, r' 2 5: A - B even. m' G 1 (mod 4). a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . even 138

3.35 The path R for Case 2.1.1, r' = 3: A - B even. m' a 1 (mod 4). a

even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.36 The path R for Cose 2.1.2.- A - B even. m' 1 (mod 4). a odd . . . . 142

3.37 The path R for Case 2.2.1, d 1 8. rt 2 7: A - B even. m' i 3

(mod 4). a even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.38 The path R for Case 2.2.1, d 2 8. r' 5 5: A - B even. m' r 3

(mod 4). a even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.39 The path R for C u e 2.2.1, d = 4: A - B even. m' i 3 (mod 4). a

even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.40 The path R for Case 2.2.2. A - B even. mt i 3 (mod 4). a odd . . . . 149

Chapter 1

Introduction

1.1 Definitions and terminology

We begin with a few basic definitions. For any terms not defined here the reader is

referred to [8].

Unless explicitly mentioned otherwise, all graphs in this thesis are simple. We

shall use h:, to denote the complete graph on n vertices and, for n even, K,, - I to denote Kn with a 1-factor I removed. & denotes the complement of K,; that

is, a graph with n vertices and no edges, and AKn stands for the multigaph on

n vertices having A edges between any two of them. We use K,,, to denote the

complete bipartite graph with the bipartition sets of sizes rn and n, and Kg(,) to

denote the complete g-partite graph with m vertices in each part.

An m-cycle, denoted by C,, is a cycle of length m; that is, a cycle with m

edges. A Hamilton cycle in a graph G with n vertices is a subgraph isomorphic to

an n-cycle.

Definition 1.1.1 We write G = HI $ H2 if G is the edge disjoint union of its

subgraphs HI and H2. If G = HI $ $ Hk, where HI, . . . , Hk are all isomorphic to

H, then G can be decomposed into subgraphs isomorphic to H; we say that G is H- decomposable and that {HI,. . . , Hk} is an H-decomposition of G. In particular, G is C,-decomposable if it can be decomposed into subgraphs isomorphic to an m-cycle.

If H is a subgaph of G and E is a subset of E(G) such that E ( H ) n E = 0 and

E ( H ) U E = E(G) , we write G = H $ El thus extending the above notation.

Definition 1.1.2 The join G w H of the graphs G and H is the graph with the

vertex set V(G Pf H) = V(G) U V ( H ) and the edge set

E(G H) = E(G) U E ( H ) U {uv : u E V(G), v E V ( H ) } .

G w H thus consists of a copy of G and a copy of H together with all possible edges

between them.

Notice that K, is isomorphic to K,,-l W K , and, for n even, h', - I is isomorphic

to (Kn-2 - I ) w ~ 2 . In these representations, the vertices of & and & are called

ccntml vertices.

Figure 1.1: A graph G and G(2).

Definition 1.1.3 Let G be a graph with vertex set V ( G ) = {zo,. . .,xk-1} and

edge set E(G). We define G(2) to be the graph with vertex set

G(2) thus consists of two copies of the graph G together with two cross edges z%x:2

and z!, x$ for every edge z;,zh of G.

Notice that Kk(2) is isomorphic to the join Kk w Kk with a 1-factor removed,

where the 1-factor consists of the edges between the corresponding vertices in the

two copies of &. In other words, Kk(2) is isomorphic to h ; k - I .

F o r a s e t S C Z w e w r i t e - S = { - s : s ~ S ) a n d , f o r a n y x ~ Z , S + x = {s+z:

s E S).

Definition 1.1.4 Let k be a positive integer and L a subset of {I, 2, . . . , 1 5 ~ ) . A circulant X = X(k; L) is a graph with vertex set V ( X ) = {uo, ul, . . . , uk-1) and

edge set E ( X ) = {&ui+l : i E Zk, 1 E L). The edge uiui+l, where I € L, is said to be

of length I , and L is called the edge length set of the circulant X. When k is even,

the edge length 5 is called the diameter length.

It is understood that the arithmetic involving the subscripts of the vertices is

carried out modulo k.

(In the literature, a circulant is more often described by its symbol S, where S = L u (-L) so that S {I,. . . , k - 1) and -S = S, rather than by its edge length

set L. However, we find the description via the edge length set more convenient for

the purposes of this thesis. )

Notice that I(, is isomorphic to X(n; (1,. . . , 9)) for n odd, and h;, - I is isomorphic to X(n; (1,. . . . f - 1)) for n even. This fact will be used extensively

throughout this thesis.

Definition 1 . l .S Let k be a positive even integer and X = X ( k ; L) a circulant with

the edge length set L. For any i E Zk, the edges uiui+l and u ~ + $ u ~ + ~ + ~ of length I are called diametrically opposed.

Definition 1.1.6 An automorphism of the graph G is a permutation 7 of the vertex

set V(G) with the property that for any two vertices u and v of G, uv is an edge of

G if and only if r ( u ) y ( v ) is an edge of G. The set of all automorphisms of G, with

the operatian of composition, is the automorphism group of G, denoted by Aut(G).

For any 7 E Aut(G), the subgroup of Aut(G) generated by y is denoted by (7); that

is, (7) = {yi : i = OJ, ... }.

Definition 1.1.7 Let G be a graph, v a vertex in G, and r a subgroup of Aut(G).

The set {y(v) : y E T), denoted by r ( v ) , is called the orbit of l? containing a.

It is not difficult to see that the orbits of r partition the vertex set V(G) and that

the vertices u and v belong to the same orbit of I' if an only if there exists an

automorphism y E r such that ~ ( u ) = u.

The action of Aut(G) on the vertex set V ( G ) induces an action on the edge

set E(G) by the rule y(uu) = y ( u ) ~ ( v ) for any y E Aut(G), uv E E(G) . For any

subroup r of Aut(G) and any edge e of G we can thus define the orbit of r containing

e by r (e ) = { ~ ( e ) : y E r). The action of Aut(G) can be further extended to subgraphs of G since for any

subgaph H of G, the graph with vertex set y(V(G)) and edge set y(E(G)), denoted

by y ( H ) , is also a subgraph of G.

Definition 1.1.8 Let X = X(k; L) be a circulant with vertex set {uo, . . . , uk-1) .

By the rotation p we mean the cyclic permutation (uo . . . ~ r - ~ ) .

It follows directly from the definition of a circulant that ( p ) = { p i : i = 0,. . . , k - 1)

is a subgroup of Aut(X) whose edge orbits are the sets {uiui+{ : i E Zk), 1 E L; that

is, the sets containing all edges of the same length.

For k even: p2 is the permutation (uo u2 . . . u ~ - ~ ) ( u ~ 143. . . u k - 1 ) SO that ( p 2 ) is

a subgroup of ( p ) with the edge orbits {uiui+, : i E Zc, i even) = (p2 ) ( ~ 0 ~ 1 ) and

{ u ~ u ~ + ~ : i E Z k , i odd) = ( ~ * ) ( U ~ U ~ + ~ ) for all I E L. Hence each set of the edges of

the same length splits into an &odd" and an "even" orbit of (d) . Both ( p ) and ( p 2 ) play a crucial role in our decompositions into m cycles.

When representing K. and K, - I as the joins w h; and (Kn-2 - I ) K*, respectively, we define p to be the permutation that acts as a rotation on the circulant

part of the graph (Kn-l or Kn,2 - I ) and leaves the central vertices fixed. In the

case of w Kl, if the vertex set of Kn-l is {uO,. . . , tln-l) and the vertex of Kl

is w , that means that p = (uo . . . u*-~)(w) In the case of (Kn-2 - I) K 2 , if the

vertex set of h;l4 - I is {uO,. . . , u ~ - ~ ) and the vertex set of R2 is { v , w ) , then

p = (uo ~ , - 3 ) ( 4 ( 4 .

A p-path is a path of length p; that is, a path with p edges and p + 1 vertices. In

a path P = sox1 . . . x,, the vertices encountered first, third, . . . (that is, xo, zz, . . = )

are called the odd vertices, and the vertices encountered second, fourth, . . . (that

is, X I , t g , . . .) are called the even vertices. The vertices xo and x, are called the

endpoints of P, xo being the initial vertex and z, being the terminal vertex. The

vertices X I , . . . ,+ I are called internal.

If P = xozl.. . x , is a ppath and Q = yoyl.. . y, is a q-path and x , = yo is

the only vertex the two paths have in common, then PQ will denote the (p + 9)-

path zozl . . . z,yl. . . y,, the concatenat ion of P and Q. Furthermore, denotes the

path z,t,-1. . . qzo, the reverse of P. Clearly, P and B represent the same graph,

however, the order in which the vertices are listed is important in concatenation.

Figure 1.2: A tig-zag path with the edge length set {al, a2,. . . ,a,] .

Definition 1.1.9 Let X = X ( k ; L) be a circulant. A path P in X with the property

that no two edges of P are of the same length is called a zipzag path. The set of

ail edge lengths represented in P is called the edge length set of P.

The name zig-zag path comes from the way such a path can be constructed from its

edge length set. It is easy to see that {al, . . . , a,}, where a1 < . . . < a,, is the edge

length set of the zig-zag path

.4lmost all cycles in our decompositions are constructed from zig-zag paths.

1.2 The origin of the problem and early results

Since an excellent survey on the problem of decomposing graphs into cycles has been

recently written by M. Anderson [3], we shall limit ourselves to a very brief history

of the problem of decomposing Kn into m-cycles. But first we would like to discuss

the necessary conditions for K, to be C,-decomposable.

Lemma 1.2.1 (Necessary Conditions for Kn to be C,-decomposable) If I\;, is C,- decomposable, then

2. n is odd, and

3. n(n - 1) = 0 (mod 2772).

PROOF. The first condition is obvious and the second condition comes from the

fact that the degree of every vertex in a cycle is 2 so that the degree of every vertex

in K, (that is, n - 1) must be even. And finally, the number of edges in Kn (that n n-l is, +) must be divisible by rn, the number of edges in an m-cycle, leading to

Condition 3. O

The question we are interested in is the following.

Question 1.2.2 Are the necessary conditions for a complete graph to admit a de-

composition into m-cycles $50 sufficient?

The origin of the problem dates back to the middle of the lgth century. In

1847, as a reply to Prize Question 1733 posed by W.S.B. Woolehouse in the Lady's

and Gentleman's Diary [29], Rev. T.P. Kirkman [14] established that the necessary

conditions for m = 3 are also sufficient. However, according to [9], he had been

preceded by Julius Pliicker, who discussed what later became known as Steiner triple

systems in his books [la] and [19] in 1835 and 1839. Steiner himself, possibly unaware

of Kirkman's work but familiar with Pliicker's books, discussed the existence of

"Steiner triple systemsn (decompositions of K, into 3-cycles) in his 1853 paper [25].

In 1892, Lucas [17] credited Walecki for settling the problem for the case m = n;

that is, the problem of a decomposition of K, into Hamilton cycles. In addition?

Walecki found a C,-decomposition of h;, - I for all even n.

The next significant contribution was to prove that for all even m and all n r 1

(mod 2m) the necessary conditions for a Cm-decomposition of K, are also sufficient.

This result was proved in a series of two papers, the first in 1965 by Iiotzig [15],

who considered the case rn I 0 (mod 4), and the second in 1966 by Rosa [21], who

considered m s 2 (mod 4).

Rosa (221 also started the work on the case rn odd, n I 1 or rn (mod 2m)

and Jackson [13] completed it in 1988 showing that the necessary conditions axe

sufficient. Since m being an odd prime power with n(n - 1) E 0 (mod 2774 implies

n E 1 or rn (mod 2m), Rosa's and Jackson's result settled the problem for m an

odd prime power.

In 1980, Alspach and Varma [2] proved that the necessary conditions are sufficient

for m being twice a prime power as well.

Several mathematicians concentrated on small values of m, some of them con-

sidering the more general problem of decomposing the complete multigraph AK,. It

was thus shown that the necessary conditions are sufficient for 4 5 rn 5 6 (Huang

and Rosa [12, 23]), m even with 8 5 m 5 16 (Bermond, Huang, and Sotteau [6]),

m odd with 5 5 m 5 13 (Bermond and Sotteau ['I]), and for m = 15 and m = 21

(Hoffman, Lindner, and Rodger [Ill). In 1991, Bell [4] extended these results to

show that the necessary conditions are sufficient for all rn < 50.

Finally, in 1997 a powerful result was obtained by Alspach and Gavlas [I]. They

proved that the necessary conditions are sufficient for all odd rn. This result was

also extended to m-cycle decompositions of K,, - I for rn and n both even. Since

this thesis stems directly from their work, we shall elaborate on their result and

outline their construction in the next section.

1.3 Recent results by Alspach and Gavlas

1.3.1 The problem

It is somewhat annoying that K,, can be decomposed into cycles only for n odd. It is

therefore natural to ask whether the complete graph on an even number of vertices

could somehow be "fixed" to admit a cycle decomposition (recall that we need the

degrees of all vertices to be even). The simplest way to achieve this one can think of

(and which preserves the simplicity of the graph) is to remove a 1-factor from A'*, thus reducing the degree of each vertex by 1. The problem we are thus facing, and

which is closely related to the problem of an m-cycle decomposition of K,,, is that

of an rn-cycle decomposition of & - I for n even.

The following are the necessary conditions for &-I to admit a C,-decomposition.

This is the analogue of Lemma 1.2.1 for n even.

Lemma 1 .&I (Necessary Conditions for h;, - I to be C,-decomposable) If h;, - I is C,-decomposable, then

2. n 2s even, and

9. n(n - 2) i 0 (mod 2m).

PROOF. The first condition is obvious and the second condition comes from the fact

that the degree of every vertex in K,, - I (that is, n - 2) must be even. Condition

3 follows from the fact that the number of edges in K, - I (that is, w) must be

divisible by m, the length of the cycle.

Question 1.2.2 can now be generalized as follows.

Question 1.3.2 Are the necessary conditions for ii, or K, - I, whichever is ap-

propriate, to admit an m-cycle decomposition also sufficient?

Alspach and Gavlas [l] have shown that the answer is &rmative whenever rn

and n are both odd or both even. The god of this thesis is to prove that the answer

to the question for the remaining two cases ( that is, for m odd, n even, and for m

even, n odd) is also ailirmative.

1.3.2 A brief outline of the constructions by Alspach and Gavlas

In order for the reader to appreciate the similarities and the differences between the

two pairs of cases, we would now like to present an outline of the constructions by

Alspach and Gavlas. The notation we shall use will be that of Section 1.1 and will

differ slightly from that of [I].

1.3.2.1 An m-cycle decomposition of K , - I for rn and n both even

The authors first prove the induction step: in a simple argument they show that it

is sufficient to consider n in the range rn 5 n c 2m. The rest of the proof splits into n n 2 odd. The first of these two cases is easy: they use two cases: even and ,,

a result by Tarsi [27] to decompose Kt into ?-paths and Hiiggkvist's Lemma [lo] n(n-2) to %ftn each :-path in K: to two m-cycles in Kn - I = 4 4 2 ) . In the case ,,

odd, K, - I is viewed as (Kn-* - I) W K~ and the construction involves two types

of cycles, which we would call central and peripheral (see Section 1.4). The central

cycles are made up of two zigzag (T-2)-paths that use diametrically opposed edges

and whose endpoints are connected to the two central verticer, to form an m-cycle

(Figure 1.3). The peripheral cycles are similar to those the reader will encounter in

this thesis.

Figure 1.3: Ezarnples of central cycles for m and n both even (left), and for m and n both odd, m 5 n < 2m (right).

1.3.2.2 An m-cycle decomposition of K, for m and n both odd

In this case Alspach and Gavlas use.the result by Hoffman, Lindner, and Rodger

[ll], which states that it is suBcient to consider n in the range m 5 n < 3m.

The construction involves two types of cycles; that is, analogues of our central

and peripheral cycles. While construction of peripheral cycles is very tedious - one has to consider the many different cases arising from the various parameters

assuming different values, usually modulo 4 - the most basic idea is always the

same: creating rn-cycles from several zig-zag paths with the same edge length set.

The central cycles, however, differ "in essence" according to whether m 5 n < 2m

or 2m 5 n < 3772. In the first case, K, is viewed as w Kl and the central

cycles are made up of two zig-zag which use diametrically opposed edges

and are connected into an m-cycle by an edge of diameter length and the two

edges from the central vertex to their remaining endpoints (Figure 1.3). In the case

2m 5 n < 3m, on the other hand, the central cycles are similar to those of the case

2m 5 n < 3771 for n odd, m even, presented in this thesis (see Figures 3.2 and 3.9

for examples).

1.4 A brief discussion of the techniques used in the constructions

In this section we would like to give a preview of the constructions for m-cycle

decompositions of K,, - I (m odd) and K, (rn even), both as a comparison to the

constructions of Alspach and Gavlas [l] and to prepare the reader for the lengthy

details that follow in the next two chapters.

1.4.1 Peripheral cycles

First we would like to explain what we mean by the term peripheral cycle.

Definition 1.4.1 Let G = X ( k ; L) be a circulant with the vertex set {uo, . , uk-I} k and the edge length set L. Denote d = gcd(k,m), n' = a, and m' = y . We are

assuming that d 1 3.

A peripheral cycle is an m-cycle C of G that fits one of the following two descrip-

tions:

1. C is the concatenation of m'-paths P, pn' (P), p 2 n ' ( ~ ) , . . . ,p(d-*'(~); or

2. C is made up of (ml- 1)-paths f7 pn' (P), $"'(P), . . . , p ( d - l ) n ' ( ~ ) together with

d linking edges.

In both cases the path P is said to genernte the peripheral cycle C.

In almost dl of our constructions the path P that generates a peripheral cycle is a

zig-zag path, the only exception being the solitary peripheral cycles of Section 3.3.2.

The term solitary will be explained shortly.

The following conditions are sufficient for a path P to generate a peripheral cycle:

In Case 1 of the definition, the sets of the internal vertices of the paths pi"'(f) , j = 0,. . . , d - 1, are mutually disjoint and the terminal vertex of P coincides with

the initial vertex of p n ' ( ~ ) ; and in Case 2, the paths pl"'(P) are vertex-disjoint.

If C is a peripheral cycle in G = X(k; L), then so are p(C) , &C), . . . , pn'-' (c). These cycles are said to be generated by the same (m' - 1)- or m'-path P as C. It

is not difficult to see that if C is generated by a zig-zag m'-path P with the edge

length set L p , then ipi(C) : i = 0,. . . , n' - 1) is a C,-decomposition of X(k, Lp). In some of our constructions, for every peripheral cycle C in the C,-decompo-

sition, @), p2(C), . . . , #-l(C) are used as well. In the cases with rn 5 n < 2772, however. n' is even and for each peripheral cycle C there are two options: either

C, p(C), . . . , #-' (c) are all used in the C,-decomposition - in that case C is

called coupled - or only C, p2 (C), . . . , pn'-2 (C) are used as such, whereas the copies

of the generating paths that appear in p(C)? P ~ ( C ) , . . . , p"'-l (C) are incorporated

into other cycles - in that case C is called solitary.

1.4.1.1 Using an auxiliary circulant to connect (mt - 1)-paths into a pe- ripheral cycle

We would now like to describe the method used to connect (m' - 1)-paths in a graph

G = X(k; L) into a peripheral m-cycle. This technique is due to Alspach and Gavlas

[I]. The assumption is that we have zig-zag (mt - 1 )-paths Pitol i = 1, . . . , c, with

the same initial vertex uo and the same terminal vertex u, (in our constructions

t E (-1, n' - 1, n' + I)) , whose edge length sets Li are pairwise disjoint. In addition,

the (m'- 1)-paths within each of the families Pi = { p ' " ' ( ~ , , ~ ) : j = 0,. . . , d - 1) are

pairwise vertex-disjoint. To choose the connecting edges we use Hamilton cycles in

an auxiliary circulant X with vertex set {vo, . . . , ud-1).

Suppose we are given a Hamilton cycle C in X . Arbitrarily orient the cycle to

obtain a Hamilton directed cycle 2. Let vj, vj2 be an edge of length j2 - jl in C. If 5 uses the zuc from vjl to vj2, connect the terminal vertex of Pi,., (that is, ~ ~ , ~ t + ~ ) with

the initial vertex of Pij2 (that is, u h , ~ ), thus using an edge of length ( jz - jl )n' - t in - G. Conversly, if C contains the arc trom v j , to vj,, connect the initial vertex of Pijl (that is, ujInt) with the terminal vertex of Pi,& (that is, U ~ , / + ~ ) , thus using an edge

of length ( j2 - jl)nf + t in G. In either case, the family 'Pi of (m'- 1)-paths together

with the linking edges arising from a directed cycle in X form an rn-cycle. Since

the two arcs corresponding to a fixed edp of the auxiliary circulant X result in two

connecting edges of distinct lengths in G, each Hamilton cycle in X can be used to

link two families of (m' - 1)-paths, one orientation for each family.

We now explain how to decompose the auxiliary circulant X into Hamilton cycles.

Let the edge length set of X be {Il, . . . , i s ) . Fint we decompose X into circulants

of degree 2 and 4; namely, circulants of the forms X ( d ; {I*}), where gcd(d, l i ) = 1,

and X ( d { l i t l j ) ) ? where the edge lengths li and l j are chosen in such a way that

the circulant is connected. A circulant of the first form is clearly a d-cycle, and a

circulant of the second form can be decomposed into two Hamilton cycles by the

following theorem by Bermond, Favaron, and Maheo [5 ] .

Theorem 1.4.2 [5] Any connected circulant of degree 4 can be decomposed into

Hamilton cycles.

Examples of circulants of degree 4 that are connected include circulants of the forms

X ( d ; {I, l + I)) , X(d; (21 - 1,21+ I)), and X(d; {21,21+ 2) ) for d odd.

We thus have a Hamilton decomposition of the auxiliary circulant X. A cycle C in this decomposition either uses a single edge length li or two distinct edge lengths

l , and l j . In the first case, the connecting edges arising from one orientation of C will all have the same length; namely, either lint - t or l i d + t. After applying

2 p, p , . . . , to the resulting m-cycle in G, all edges of this length will be covered.

We thus have a choice of either using both or just one of the two orientations of

C, thus linking up one or two families of (m' - 1)-paths. If C uses two distinct

edge lengths li and lj, however, a single orientation of C may produce connecting

edges of four different edge lengths; namely, lint - t, lint + t , ijnf - t , and ljn' + t. In order to account for all the edges of these lengths in G, we therefore must use

both orientations of C. A circulant X(d; {li,lj)) is thus used to link four families

of (m' - 1)-paths. It is now easy to see that, provided that a s d c i e n t number of

suitable edge lengths ll , . . . , lb can be found, any number of families of (m' - 1 )-paths

can be linked into m-cycles. We shall give the details of how to choose the edge

lengths in X and how to decompose X into circulants*of degree 2 and 4 for each of

the constructions that uses this method when needed.

1.4.2 Diameter cycles and central cycles

We are hoping that by now the reader has a fairly good "feelingn for peripheral

cycles since we are about to introduce the other two types of cycles.

Definition 1.4.3 Let k be an even positive integer and let G = X ( k : L) be a circu-

lant with the edge length set L, where 1 E L. A diameter cycle CD is an m-cycle in

G consisting of two zig-zag (7 - 1)-paths P and $(P) together with two edges of

diameter length $. The zig-zag ( y - 1)-path P is said to generate the diameter cycle CD.

For an example of a diameter cycle see Figure 3.21.

Definition 1.4.4 An rn-cycle in K , or Kn - I that is neither a peripheral cycle nor

a diameter cycle is called cent~al.

As we shall see, the construction of central cycles greatly varies from case to case - hence the open-ended definition. The simplicity of the central cycles of Alspach and

Gavlas is due to the matched parities of the parameters rn and n; that, of course, is

no longer the case in our problem.

1.4.3 A brief outline of the constructions

We are now ready to present a brief outline of our constructions.

1 . 4 1 An rn-cycle decomposition of K, - I for m odd, n even

In this case we first prove the induction step; that is, that we can limit ourselves to

the interval m 5 n < 3m, which i s the analogue of the result by Hoffman, Lindner,

and Radger [Ill. The rest of the proof splits into two cases: 2m 5 n < 3m and t n s n < 2 r n .

For n in the range 2m n < 3m, Kn - I is viewed as Kf (2). We first decompose

Kg into m-cycles and rn-paths and then show that each of these can be "lifted" to

four rn-cycles in Kt (2).

For n in the range rn 5 n < 2m, K,, - I is viewed as (Kn-? - I ) R2. The construction involves two types of cycles: peripheral (coupled and solitary) in

K, ,4 -I, and central. The starter central cycle C is made up of two paths R1 and Rz which are connected by the edges from their endpoints to the two central vertices

(that is, the vertices of ~ 2 ) to form an m-cycle. The paths R1 and R2 together

consist of two copies of one or two zig-zag paths (where the two copies belong to

distinct orbits of ( p 2 ) ) together with a copy of the zig-zag path that generates the

solitary peripheral cycles (see Figures 2.9 to 2.16). All other central cycles are then

obtained from C by applying the rotations p2, p4,. . . , pnm4.

1.4.3.2 An m-cycle decomposition of K, for m even, n odd

The induction step for this case is simple and has been obtained by Rodger [20]. Again, it states that it is sufficient to consider n in the range m < n < 3m, and the

construction splits into the same two large cases as before.

In the case 2m < n < 3772, no central vertices are used. The construction

involves two types of cycles: central and peripheral. The starter central cycle is

simply a zig-zag m-path with the initial and the terminal vertex coinciding (see

Figures 3.2 and 3.9 for examples). A 1 other central cycles are obtained from C by

applying the rotations p, p 2 , . . . pn-l. The majority of the work for this case lies in

the construction of peripheral cycles, which are made to fit the conditions for the

existence of the central cycles described.

In the case rn 5 n < 2m, Kn is viewed as Knml WI K l . Two subcases have to be n-m considered according to whether c = is less than $ = gcd(m, n - 1) or not.

For c 2 $ (roughly speaking, when the number of m-cycles is relatively large), the

construction involves three types of cycles: peripheral cycles (solitary and coupled,

or just coupled) and diameter cycles in K,,+ and central cycles, which pick up the

central vertex (the vertex of Kl). The starter diameter cycle CD is made up of

two zigzag ( y - 1)-paths P and p q ( ~ ) together with two edges of the diameter

length, which belong to distinct orbits of (p2) (see Figure 3.21). All other diameter 2 4 cycles are obtained from CD by applying the rotations p , p , . . . , py-2 (note that

9 is even). The starter central cycle C consists of a copy of the zig-zag path

that generates the solitary peripheral cycles (if any), a copy of the zigzag path that

generates the diameter cycles, two copies, belonging to distinct orbits of ($), of a

zig-zag path that uses the remaining edge lengths, and two edges from the central

vertex (see Figures 3.22 to 3.40). The other central cycles are obtained from C by

applying the rotations p2, p4, . . . , pn-3.

For c < $ (that is, when the number of rn-cycles is relatively small), only two

types of cycles are used; namely, peripheral (solitary and coupled) in and

central, which pick up the central vertex. We have two starter central cycles Cl and C2. Cl consists of a path P (two diametrically opposed zig-zag paths connected by

an edge of diameter length), 3 or 5 linking edges, and two edges from the central

vertex. C2 consists of the path p(P) and, again, 3 or 5 linking edges, and two edges

from the central vertex (see Figures 3.10 and 3.11). The other central cycles are 2 4 obtained from C1 and C2 by applying p , p , . . . , p?-2. The solitary peripheral

cycles require a special construction to accomodate the "3 or 5" linking edges in the

central cycles (see Figures 3.13 to 3.20).

Chapter 2

Decomposition of K, - I into m-cycles, where n is even and m is odd

2.1 Inductive step and the main result

2.1.1 Introduction

In their 1989 paper 'On the construction of odd cycle systemsn [ll], Hoffman,

Lindner, and Rodger proved that for m and n both odd, if K. is C,-decomposable

for all n in the range m 5 n < 3m that satisfy the necessary condition n(n - 1) a 0

(mod 2 4 , then h;, is C,-decomposable for all n satisfying n(n - 1 ) r 0 (mod 2m).

We use the technique of their proof to prove the equivalent statement for n even:

Theorem 2.1.1 (Induction Theorem) For m odd and n even, if K , - I is C,- decomposable for all n in the range rn 5 n < 3m that sat* the necessary condition

n(n-2) 0 (mod 2 4 , then Kn -I is Cm-decomposable for dl n satisfying n(n-2) I

0 (mod 2m).

First we introduce the terminology and notation, which differs from that of 1111.

Let m = 21 + 1 be a fixed odd integer. Recall that G M Kt is the join of the

graph G with the complement of the complete graph on t vertices, and that G w KO is simply G. Recall that K42) is the join Km w K, with a 1-factor removed, where

the 1-factor consists of the edges between the two copies of a vertex in K,. We shdl

view the graph K2rn - I as Km(2) with vertex set VoUK, where Vi = {(uy) : j E 2,). (1) (1) An edge u ~ ) u ~ ~ ' (respectively u,, uj2 ) is called an edge of It$ (respectively right)

p u n length jz - jl or jl - j2, whichever is in the set L = {1,2, . . . , I ) . An edge

u!)uj;') is called an edge of mized length j2 - jl, where jz - jl E 2, - (0).

If A L, C L and B 2, - {0), define Km(2)(A, B, C) to be the spanning

subgaph of &(2) whose edges are all those with left pure length in A, or mixed

length in B, or right pure length in C. Note that I(,(Z)(A, B , C ) is isomorphic

to Km (2) (C, - B, A) , as we11 as to K, (2) (A, B + z, C) for any x E 2, such that

0 @ B + x . As already mentioned, we label the vertices of Kz, - I in such a way

that K2, - I = Km(2)(L, 2, - {0), L). To prove Theorem 2.1.1, we first need to show that the graphs Km(2) w Kt are

C,-decomposable for the various (and small enough) values of t.

The following lemma, an adaptation of a lemma by Stern and Lenz [26, 111, will

be used in the proof of the main result of the next section.

Lemma 2.1.2 [26, 111 Let G be a regular graph of degree d and H = G x K2 , that is, a graph with wertez set V(G) x & and with the vertices (u , i ) and ( v , j ) adjacent

if and only if either u = v , or i = j and uv E E(G). Then H is properly d + 1 edge

coloumble and each colour class is a i-factor in H.

PROOF. By Vizing's theorem [28] the edges of G can be properly coloured with

d + 1 colours. Let f : E(G) -, {1,2,. . . , d + 1) be a proper edge colouring of G with colours 1,2,. . . , d + 1. We extend f to a proper d + l edge colouring g of

H as follows. If uv E E(G), let g((u,O)(v,O)) = g ( ( u , l ) ( v , l ) ) = f(uu). For every

v E V ( G ) , there is exactly one colour c (v ) not represented by the edges of G incident

with v . Let g ( ( v , O)(v, 1)) = c(v) .

Since each of the d + 1 colour classes is a matching in H, it contains at most

I V(G) I edges. On the other hand, the union of the d + 1 colour classes is E (H) with

cardinality (d + l)IV(G)(. Hence each colour clam is in fact a 1-factor. 0

2.1.2 Some C,-decomposable graphs

In this section we'll show that certain graphs Km(2)(A, B, C) w K$ are C,-decom-

posable as in [Ill. The vertices of K~ are denoted by w l , . . . , w,.

Lemma 2.1.3 (Type 1) If s is an odd integer. 1 < s < I . then h',(2)(0. {I&. . . . m - 2s),0) & is C, -decomposable.

4 -3 -2 - 1 0 I 2 3 4

Figure 2.1: Lemma 2.1.9: Co in the Cs-decomposition o f&(2) (0 , {1,2,. . . ,7),0) w

PROOF. First, let C be the (m - $)-cycle

Notice that the first rn -- 2s edges use each of the mixed lengths 1,. . . , m - 2s

precisely once. Let the last 3 edges be ziyil i = 1, . . . , s. Replace the edge ziyi in C by the 2-path xiwiyi, thus obtaining an m-cycle Co. Finally, for j = 1,. . . , m - 1, obtain Cj from Co by adding j to the subscripts of the vertices other than the w;. It

is not difficult to see that the m-cycles Cj are pairwise edge disjoint and that they

use up all the edges of K,,,(2)(0, (1'2,. . . , m - Zs), 0) w K ~ . 0

Lemma 2.1.4 (Type 2) If s is an even integer, 0 5 s 5 I, then Km(2)({1), {1,2,

. . . ,m - 2s - I),@) w K~ is C,-decomposable.

Figure 2.2: Lemma 2.1.4: Co in the Cs-decomposition of K9(2)({4), { l , 2,3,4),0) w &.

PROOF. Let C be the (m - +cycle

We proceed as in the previous lemma. The first m - 28 - 1 edges of mixed lengths

use each of the mixed lengths 1, . . . , m - 2s - 1 precisely once. Let the last s edges

be xiyi, i = 1, . . . , s, and replace the edge xiyi in C by the 2-path xiwiyi to obtain

the m-cycle Co. Finally, for j = 1, . . . , m - 1 , obtain Cj from Co by adding j to the

subscripts of all vertices of Co except for the wi. The m-cycles Cj then represent a

decomposition of the graph Km(2)({1) , {1,2,. . . ,m - 29 - 1),0) w c. 0

Lemma 2.1.5 (Type3) If S E (2k- 1 : 1 5 k 5 I ) , then G = Km(2)({I) ,S~(S+ I), 0 ) is C,-decomposable.

PROOF. For each 2k - 1 E S we take two m-cycles:

Figure 2.3: Lemma 2.1.5: Cl and C; in the Cg-decomposition of K9(2)({4)1{1,2) , 0)

Since Ck and Ci are vertex-disjoint on & and since they use distinct edges of pure

length 1, Ck and Ci are edge-disjoint. They use 21 edges of each of the mixed lengths

2k - 1 and 2k among them. The two remaining edges of mixed length 2k - 1 or 2k 0) ( 0 ) are UZ+~ u& and u,,, u-,-,, which snugly fit into the gap in the m-cycle of the

edges of pure left length 1 caused by the removal of the two edges ufi+, u(_ql and (0 ) (0) (0 ) ( 0 ) ( 0 ) u-k u-,-,, which were used in Ct and Ci. Since the 2-paths u,,, + I u+ u - , ~ -, and (0) (0) (0) u - , ~ + ~ u-k2 u + - ~ are edge disjoint for kl # kz (recall that 1 5 kl , k2 5 I ) , the edges

remaining after the removal of the Ck and Ci for dl %k - 1 E S will necessarily form

an m-cycle. This additional rn-cycle together with all the Ck and Ci thus form a

C,-decomposition of the graph G. O

Lemma 2.1.6 (Type 4) Let each of the sets A and B be either L = {I,. . . , I ) or

L - 1 . Then K,,, (2) (A, 0, B) is C, -decomposabk.

PROOF. Since X(m; { l ) ) is an m-cycle and X(m; L - {I)) can be decomposed into

~raphs X(m: {1,2)),. . . ,X(m; (1-2,l-1)) or X(m; {l}),X(m; {2,3)), . .. ,X(m; { l - 2 , l - I)), depending on whether 1 is odd or even, and since these graphs are C,-

decomposable by Theorem 1.4.2, the result follows. 0

For 1 5 i 5 4, we say that a graph G is of type i if it is isomorphic to the

C,-decomposable graph of Lemma 2.l.(i+2).

Lemma 2.1.7 The graph K2, - I is Cm-decomposab1e.

PROOF. Write &, - I as Km(2)(L, 2, - {0), Lj and decompose the latter into

Km (2) ( { l ) , 2, - (01, a), which is of type 3, and K , (2) (L - { 1 ) , O , L), which is of

type 4. 0

The following lemma will be a major building block in the proof of the Induction

Theorem 2.1.1.

Lemma 2.1.8 Let t be a positive even integer. If t 5 *, then (hi, - I ) Kt b Cm -decompossble.

PROOF. First write t = ql + r , where 1 5 r 5 I . Since't is even and t 5 = 212 + 21 + 1, t 5 (21 + 1)1+ 1. Hence q 5 21 + 1 < - m + 2r - 2.

Recall that Kg, - I is isomorphic to h',(2)(L, Zm - {O), L ) . We thus need to

show that = &(2)(L, Z, - { 0 ) , L) w K~ is C,-decomposable.

CASE 1. 1 is even (and hence r is even).

First decompose G = G1 $ (G2 W & I ) , where

and

G, = Km (2) (6, {rn - 2 t , . . . , m - 11, {I)).

GI is of type 2 and thus C,-decomposable.

G3 and G4 are both C,-decomposable since G3 is of type 4 and GI is of type

3. There are 2(1- 1) + (27 + 1) = m + 2r - 2 cycles in this decomposition of G2.

Since q 5 rn + 2r - 2, we can choose q of these cycles GI, . . . , C, and let their union

be the graph Gs. Let the union of the remaining rn + 2r - 2 - q m-cycles be Gs. We thus have G2 = G5 $ Gs with G6 already decomposed, and it only remains to

decompose GS w xql. Now Gs K~~ can be decomposed into q graphs isomorphic

to Km(2) ( ( 1 ) : 0,0) cu i?! (each corresponding to one of the Clt.. . , C,): which is of

type 2 and thus C,-decomposable.

CASE 2. 1 and r are both odd (and hence q is odd).

First decompose G = 4 $ (G2 Kql) , where

and

G2 = Km(2)(L, {m - 2r + 1,. . . , m - I ) , L).

GI i s of type 1 and thus C,-decomposable. It thus remains to decompose G2 w Kqr. Two subcases arise depending on the value of q.

SUBCASE 2.1. q < 2r - 1. We decompose G2 = G3 @ G4 $ GS, where

GJ and Gs are of types 4 and 3, respectively, and thus C,-decomposable. On the other hand, G4 BJ $1 can be decomposed into q graphs isomorphic to Km(2)(0, {1},0)

M Kl, which is of type 1.

SUBCASE 2.2. q 2 2r + 1. First decompose G2 = G3 @ G4, where

and

G4 = Km(2)(0, { m - 2r + I , . . . , m - 1},0).

G3 is of type 4 and thus C,-decomposable. We may assume that the decomposition

is symmetrical in the sense that if ujp) . . . ui:) u,!? is an rn-cycle in the decomposi-

tion, then so is u!:). . . u!? u!:). Choose f (* - 2r + 1) cycles on the "leftn and the

corresponding f (q - 2r + 1) cycles on the "right" and let their union be the graph

Gs . Now Gs $ K, (2) (0, { m - 2r + 1 }, 0) can be decomposed into q - 2r + 2 1-factors

by Lemma 2.1.2 and Km(2)(0, {m - 2r + 2,. . . , m - I) ,@) is decomposed into 2r - 2

1-factors; that is, q 1-factors altogether. Hence (Gs $ GI) w ~ 9 1 is decomposed into

q gaphs isomorphic to K , (2) (0, {I), 0) KI, which are of type 1.

CASE 3. 1 is odd and r is even (and hence q is even).

Decompose G = GI $ (GI K ~ ~ ) , where

and

G2 = L ( 2 ) ( L - { l ) , {m - Z T , . . . , rn - I), L).

4 is of type 2 and thus C,-decomposable. To decompose G2 K ~ ~ , we again have

to consider two subcases with respect to q.

SW BCASE 3.1. q 5 2r. Decompose G2 = Gg $ GI $ G5, where

Gs = K , (2)(0, { m - 2+, . . . , m - 1 - q ) , { I ) )

is of type 4, and

Gs = K,,,(2)(0, {m - q,. . . , m - 1},0).

Now CS M Kqr can be decomposed into q gaphs isomorphic to Km(2)(0, {I), 0) w &, which are of type 1.

SUBCASE 3.2. q 2 2r + 2. First decompose G2 = G3 $ G4 $GS, where

is an rn-cycle,

and

G4 = K m ( 2 ) ( L - {I), 0 , L - {I)),

As in the case q > 2r + 1 with r odd, decompose G4 = Km(2)(L - {1) ,0, L - { I ) ) "symmetricaI1y"and then choose f (q - 2r) of these m-cycles on the "leftn and the

corresponding i ( q - 2 r ) cycles on the "rightn to form the graph Gs. Now Gs $ Ge can be decomposed into q 1-factors using Lemma 2.1.2 and thus (G5 $ Gs) w is

decomposed into q graphs isomorphic to h', (2) (6, { I ) , 0) w K ~ , which is of type 1.

This completes the proof.

Corollary 2.1.9 K2*+2 - I is C,-decomposable.

PROOF. Kzm+z - I is isomorphic to (&, - I ) W K2. Since t = 2 5 9 for all m 2 3, the graph is C,-decomposable by Lemma 2.1.8. 0

We adopt the next theorem from [ll] without proof and then we are ready to

prove Theorem 2.1.1.

Theorem 2.1.10 [Ill if g >, 3, the complete g-partite gmph &(2,) is C,-decom-

posa ble.

2.1.3 Proof of the Induction Theorem 2.1.1

PROOF. Since nl 3 nz (mod 2m) implies nl(nl - 2) = nz(nz - 2) (mod 2m), it is sufficient to prove the following: if K,, - I is C,-decomposable for rn 5 n < 3m, then K,,+2,, - I is C,-decomposable for all g 2 1.

If m 2 7, n < 3m implies n < q. We can thus use Lemma 2.1.8 to decompose

K,,+a, - I as follows. If g 2 3, Kn+2mg - I is decomposed into the complete

g-partite graph Kg(2m), g copies of (4, - I) w Kn, and K,, - I , which are C,- decomposable by Theorem 2.1.10, Lemma 2.1.8, and the assumption, respectively.

If g = 1 , Kn+2m - I is decomposed into (Kzm - I) w K,, and h;, - I, which are

both C,-decomposable. If g = 2, we decompose K,,+4, - I into ( K 2 , - I ) &+,, and Kn+2rn - I. The latter is C,-decomposable by the previous observation and

the former is Cm-decomposable by Lemma 2.1.8, but as far as we know, this can be

done only for n + 2m 5 e; that is, for rn 2 11. We thus need to consider the case m 9, which means that rn is a prime power.

Thus n(n - 2) E 0 (mod 2772) and m 5 n < 3m imply either n = 2m or n = 2m + 2.

Hence n 5 for rn 2 5, so that Kn+2rng - I is Cm-decomposable for all g except

maybe for g = 2. If n = 2m, Kn+,rn - I = Ksm - I, which is decomposable into

K3(2m) and three copies of Kz, - I, and these graphs are Cm-decomposable by

Theorem 2.1.10 and Lemma 2.1.7. If n = 2m + 2, Kn+4, - I = K2+6rn - I , which

is decomposable into K3(?rn) and three copies of (Kzm - I) w x2, the latter being

C,-decomposable by Lemma 2.1.8.

The remaining case is thus m = 3. Fint let n = 2772. K2,, - I is Cm- decomposable by the assumption and by Theorem 2.1.10 whenever g # 2. A C3- decomposition of h;, -I = KI2 - I is found as follows. Consider a C3-decomposition

of KI3. Remove a vertex v from KIS together with the six 3-cycles that contain v .

Since the six edges in these 3-cycles that are not incident with v are independent,

the remaining 3-cycles represent a C3-decomposition of KI2 - I . Finally, let n = 2m+2 with m = 3. For g 2 2, Kn+2rn9 - I = (Kz,.(g+ll- I ) w K2

is decomposable into the complete (g+l)-partite gaph and g + 1 copies of

(K2m - I) W K ~ , ail Cm-decomposable. The remaining case is KI4 - I. First,

decompose it into K6 - 1 and (& - I ) W G. The first gaph is C,-decomposable

by Lemma 2.1.7. To decompose (& - I) & , choose a 1-factorization { 4, . . . , Fs) (4 (i) of & - I . If we denote the additional 6 vertices by wl , . . . , w~ and let F, = { x j yj .

j = 1, . . . ,4), then { ~ i z ~ ) y ~ i ) w i : i = 1, . . . ,6, j = 1, . . . ,4) is a decomposition of

(& - I) w & into Scycles.

This completes the proof. 0

2.1.4 The main result

Theorem 2.1.1 tells us that it is sufficient to find C,-decompositions of K, - I for

dl n in the range m 5 n < 3m that satisfy the necessary condition n(n - 2) = 0

(mod 2m) and this is our goal for the rest of the chapter. The following theorem,

which is the main result of this chapter, then draws the conclusion.

Theorem 2.1.11 Let n be an even integer and m be an odd integer such that 3 5 m 5 n and n(n - 2) m 0 (mod 2m). Then h:, - I is C,-decomposable.

PROOF. By Lemma 2.3.5 (to be proved later), Kn - I is Cm-decompsable for all n in

the range m 5 n c 2m that satisfy the necessary condition n(n - 2) r 0 (mod 2m),

and by Lemma 2.2.4, K. -I is C,-decomposable for all n in the range 2m 5 n < 3m

that satisfy the necessary condition. It now follows from the Induct ion Theorem 2.1.1

that Kn - I is Cm-decomposable for all n 2 m that satisfy the necessary condition

n(n - 2) 1 0 (mod 2m). 0

2.2 C,-decomposition of K, - I for n even, m odd, and 2msn<3m

Throughout this section, let k = 5. As we shall see, it is convenient to view the

graph K , - I as Kk(2).

The basic idea of this construction is very simple. First we decompose Kk into

rn-cycles and m-paths, and then we carry this decomposition over to Kk(2) . The

next lemma describes how to construct the m-paths in Kk and how these give rise

to m-cycles in Kk(2). Lemma 2.2.2 explains how the m-cycles in Kk are carried over

to form rn-cycles in &(%).

Lemma 2.2.1 Let k be even and m 2 5 . Furthemon, let X = X(k; S,) be a

circulant with the edge length set S, = {al, dl , . . . , a m - & 1) such that a1 c a2 < T' 2 k k . . . < a- < 5 and 5 - a1 E Sp.

Then X ( 2 ) is C, -decomposable.

PROOF. First we decompose X into m-paths. Let the vertex set of the circulant X be {uo,ul ,.... ur-1). Find r, 1 < r < T, such that 5 k - a1 = a,. Let h =

and A = a1 - a2 + + (-l) '~,-~ + (-l)'+'a,. We define the following paths:

Notice that the path R uses each of the lengths a2, ..., a,- 1 precisely once, while

.. the path Q uses each of the lengths a ,+ l , . ,ah precisely twice aad the diameter

length exactly once.

Figure 2.4: m-paths Po and PO.

Now we define the m-paths Po and as follows:

It is easy to see that each of the rn-paths Po and PO contains a pair of diametrically

opposed edges of each of the lengths al , . . . , a h together with one edge of the diameter

length 5 . Each of the families

and

thus represents a decomposition of the circulant X into rn-paths. Moreover, T : P - P, T(pi(Po)) = &PO) , is a bijection.

Unlike in the case of an m-cycle (Lemma 2.2.2), for an rn-path P, P(2) is not C,-decomposable. However, each P E P gives rise to a gaph G 2 ( P ) , closely related

to P ( 2 ) , which is C,-decomposable and which we now define.

Let

P = X o X l . . XrZr+lXr+2.. . X t - 2 X t - l X l . . Xm-lXm

be an m-path in P and

be the corresponding m-path in P. We define G2(P) to be the gaph obtained from

P(2) by replacing each of the two copies of P by T (P). That is, G 2 ( P ) is the gaph

with the vertex set

and the edge set

E(G2(P)) = {4 ,z i2 : zirzi E E(T(P) ) , j E 22) U { ~ ~ , + ~ ~ j : =il*i2 E E ( P ) , j E Z1).

Figure 2.5: m-paths P and T(P ) , and a C,-decomposition of G2(P).

The four m-cycles in the decomposition of G 2 ( P ) are:

It remains to show that G2 = {G2(P) : P E P) is a decomposition of X ( 2 ) . Take an edge e = ui:u$ E E(X(2) ) . If jl = j2, since u;,ui2 lies in a unique P E P, e

lies in a unique member of &, namely G2(TD1(P) ) . If jl # j2, then ui1 s2 lies in

a unique PI E P and in a unique P2 E P. If the length of ui,ui2 is neither a1 nor

- a*, then P2 = T ( P I ) and hence e lies in a unique member of &, namely G2 ( P I ) . 2

If. however, the length of ui, ui2 is either a: or - al, then P2 # T ( P l ) . In this case,

by the definition of G 2 ( P ) , e lies in G4Pl ) but not in G2(P1(P2)). This completes the proof. 0

Lemma 2.2.2 Let G be a gmph. If G is C,-decomposable, then G(2) is C,- decomposable.

PROOF. It suffices to show that for an m-cycle C, C(2) is C,-decomposable. As in

the rest of this section, we assume that m is odd. However, the statement is also

true and, in fact, obvious for rn even.

Figure 2.6: A C,-decomposition of Cm(2) form = 3 and m 2 3.

Let C = 10~1.. . x,-~zo. Then E ( C ( 2 ) ) = {tihgl : i E Z,, jl, j2 E &}. The four m-cycles in the decomposition of C(2) are:

Recall that k = t . We now assume that k is even. Let's have a look at the

parameters at play. We let n = 2m + r , where r < rn and r is even. Let d = gcd(m,n) = gcd(rn, r) and denote k = k'd, rn = m'd and r = t'd. If d = 1, since

n(n - 2) 0 (mod 2m), n - 2 r 0 (mod m). Since 2m 5 n < 3m, n = 2m + 2, a case settled by Corollary 2.1.9. We may thus assume that d 2 3. Similarly, if m' = 1, n = 2m, a case settled by Lemma 2.1.7, so we may assume that m' 2 3.

Since 4k'd(k - 1 ) 0 (mod 2mtd), k - 1 E O (mod m'). Let k - 1 = h'. Since

r < m - 1, bm' = k - 1 = m + 5 - 1 5 9. It follows that b y. Since d is odd, 3d 1 b l * .

We would now like to decompose Kk into m-paths and m-cycles. The number

of edge lengths used in the m-paths from Lemma 2.2.1 is F. This leaves

edge legths to be used in m-cycles. Since b and d are both odd, c = is an

integer. As we shall see in the next lemma, this allows for a construction of c

families of m-cycles, each using rn' distinct edge lengths. 36 1 Notice that, since b 5 5,

Lemma 2.2.3 Let k be even, m', k', and c as defined above. There ezzsts a subset

S, of the edge length set {1,2,. . . ,! - 1) with the following properties:

1. 1 E Sc and 2 6 S,,

PROOF. We prove the lemma by constructing c families of peripheral cycles.

First assume rn' 2 5. Define a zig-zag (m' - 1)-path

Notice that the edge length set of Po,o is

Figure 2.7: Lemma 2.2.9: the rig-zag (m' - 1)-path Poqo for m' 2 5.

&+a = mt 5 kf-1, For j = 1, . . . , d- 1, let Poj = p t " ( ~ ~ , ~ ) . Notice that, since 2

the (m' - 1)-paths Po,. axe pairwise vertex-disjoint.

For i = I , . . . , c - 1 , obtain the kg-zag (m' - 1)-pathPiVo from Po,o by adding 22k' to the subscripts of the even vertices. That is, let

As before, let Rj = p'k'(P;:,o) for j = 1,. . . ,d - 1. For i = O , l , . . . , e - 1, denote

Pi = {Pi,j : j = 0,. . . , d - 1). The set of edge lengths used by the family of

(m' - 1)-paths Pi is thus Li = Lo + 2ikf .

We now use the technique of Paragraph 1.4.1.1 to connect each family of (m' - 1)-

paths into an m-cycle using an auxiliary circulant X = X(d; {l, 2,1,6,. . . , 2 Lf 1 }). If c is even, X is first decomposed into circulants X(d; {I)), X(d; {2}), X(d; (4, ti)), . . . , X(d; {c - 2 , c h or X(4 t w , W d ; {2)), X(d; {4))? X(d; (6, B)), A d ; {c - 2, c)),

depending on whether f is odd or even. Since d is odd, the circulants X(d; {22,2i+2))

are connected and hence decomposable into Hamilton cycles by Theorem 1 A.2. We thus have a Hamilton decomposition of X. Since the terminal vertex of each of the

paths Er is UV-1, an edge of length I in a d-cycle C gives rise to a connecting edge of length (1 - 1) k' + 1 or (1 + 1)k' - 1, depending on the orientation of C we are

using as explained in Paragraph 1.4.1.1. The d-cycles X(d; {I)) and X(d; {2}) will

be used with only one orientation each; namely, the orientation that gives rise to

connecting edges of length 1 and kt + 1, respectively, whereas X ( d ; (4)) is used with

both orientations, thus giving rise to connecting edges of lengths 3kt + 1 and 5 k t - 1.

For the circularits X(d: {22,22 + 2)) we don't have a choice; each is used to link up 4

families of (m' - 1)-paths, giving rise to connecting edges of lengths (22 - 1)k' + 1, (22 + l)k' - 1, (22 + 1)k' + 1, and (22 + 3) kt - 1. The set of lengths of the connecting

edges is thus

If c is odd, X is first decomposed into X(d; {I)), X(d; {2,4)), . . . , X(d; { c - 3, c - I}), or X(d; {I)), X(d; {2}), X(d; {4,6)), . . . , X(d; { c - 3, c - I)) , depending on

whether 9 is even or odd. The d-cycle X ( d ; {I)) will be used only with the orien-

tation that gives rise to connecting edges of length 1, whereas the d-cycle X(d; (2)) is used with both orientations. The set of lengths of the connecting edges is thus

Notice that in both cases Lx and u::; Li are disjoint. Let S, = (U:;: Li) U L x . If Ci is the m-cycle arising from the family Pi of (m' - 1)-paths, then

is a C,-decomposition of the circulant X(k; S,). The longest edge in X ( k ; S,) has

length rnt + (2c - l ) k t . Since c 5 9, C' 2 m'+ 1, and mt 2 5 ,

It is now easy to see that the set S, satisfies Conditions 1 - 4 of the lemma.

Now let rnt = 3. Hence nt = 4. Let

and obtain Pia from PIlo by adding 2(i - 1)k' to the subscript of the second vertex.

The edge length sets of the zig-zag paths Poto and PiVo are thus

for i = I,. . . , c - 1. Obtain the 2-paths Pi,jl j = 1,. . . , d - 1, from Pi,o as before

and use the same method to link them into an m-cycle C,. Again, the sets Lx and

Ufzi Li are disjoint SO that

is a C,-decomposition of X ( k ; S,), where S, = (Uzli L, ) u L x . Since c < 9 and

k' = 4, the maximum element of S, is

It is now clear that the set S, satisfies Conditions 1 - 4.

We are now ready to prove the main result of this section.

Lemma 2.2.4 Let n be an even integer and rn be an odd integer such that 6 5 2m 5 n < 3m and n(n - 2) 0 (mod 2m). Then K, - I is C,-decomposable.

PROOF. K, - I is isomorphic to Kk(2), where k = B. Notice that n(n - 2) = 0

(mod 2m) implies k(k - 1) r 0 (mod 2m).

If k is odd, since k(k - 1) r 0 (mod 2m), Kk is C,-decomposable by the result

of Alspach and Gavlas [l]. Hence Kk(2) is C,-decomposable by Lemma 2.2.2. We may now assume that k is even. Let d, kt, m', r', and c be defined as in

the discussion on page 31. We have seen that the cases d = 1 and m' = 1 have

already been settled by Corollary 2.1.9 and Lemma 2.1.7, respectively, so we may

assume that d 2 3 and rn' 2 3. Let S, be the set from Lemma 2.2.3. We thus

have a C,-decomposition of X ( k ; S,). As shown in (2. I), S, = (1,. . . , 4 ) - S, has cardinality as required by Lemma 2.2.1. Moreover, a1 = 2 and the longest edge

in X ( k ; S,) has length less than 5 - 2. The conditions 4 - a1 E Sp and E S, of

Lemma 2.2.1 are thus satisfied.

We have thus decomposed Kk into X(k; S,) and X(k; S,). Since X(k; SJ(2) is Cm- decomposable by Lemma 2.2.2 and X ( k ; Sp)(2) is C,-decomposable by Lemma 2.2.1,

Kk(2) is C,-decomposable. 0

2.3 C,-decomposition of K, - I for n even, m odd, a n d m < n < 2 m

2.3.1 Preliminaries

Throughout this section it is assumed that n is an even integer and m is an odd

integer such that 3 < m 5 n < 2m and n(n - 2) r 0 (mod 2 4 .

We shall view the graph h;, minus a 1-factor as the join (KnW2 - I) W K~ where

the vertex set of Kn-2 is {uO, ~ 1 , . . . , ~ n - 3 1 , the vertex set of K 2 is { v , w ) , and the

1-factor of K, consists of all the edges of diameter length in Kn-2 together with

the edge uw. We refer to v and w as the central vertices and denote the diameter

length 9 by D. Let us now discuss the parameters. The remainder r = n - m is clearly odd and

r S rn - 2. Let d = gcd(m,n - 2). Then d is odd and d = gcd(rn, r - 2) as well.

Denote n - 2 = dn', rn = dm', and r - 2 = dr'. We thus have D = y. The necessary condition 2m(n(n - 2) and the restriction m 5 n < 2m have the

following implications. If d = 1, then mln and hence rn = n, a contradiction since n

is even and m is odd. Hence d 2 3. If m' = 1, then m((n - 2) and hence m = n - 2,

which is again a contradiction. Hence m' 2 3.

Now if K, - I is C,-decomposable, the number of m-cycles is going to be

T(? -21 which is assumed to be an integer. We thus find that must be an odd integer.

Hence there exists an odd integer c such that r = em'. Notice that, since r m - 2, m' 5 dmr- 2, whence c 5 d - 3. Since c and d are both odd integers, c 5 d - 2.

The expression (2.3) for the number of m-cycles thus attains the form

This suggests the following scheme: central cycles would be generated by apply-

-- "-' 1, to a central cycle that contains both centrai ing the rotation p2i, I = 0, . . . , vertices; this would take care of all the edges between Kn-2 and ~ 2 . In addition,

c$ peripheral cycles would be found by applying p", i = 0, . . . , $ - 1, to a solitary

peripheral cycle, and by applying $, i = 0, . . . , n' - 1, to each of the 9 coupled

peripheral cycles.

Observe that, since n - 2 is even, the permutation group ( p 2 ) has two orbits on

the set of the edges of a fixed length as explained on page 4.

The details of the construction of central cycles depend on whether r' = 1 or

r 3. The following lemma, however, provides the basis for the construction of

central cycles in both cases.

Lemma 2.5.1 Let Lo and LC be two disjoint aubsets of the edge length set S = (1, . . . , D - 1 ) and let P be a zig-zag mf-path with the edge length set Lo. Fur-

thennore, let Rt and R2 be two vertez-disjoint paths in K .4 - I with the followzng properties:

1. the length of every edge of R1 $ R2 i9 in Lo u LC,

2. R1 $ R2 contains ezactly one edge of each of the lengths in Lo and this edge

belong8 to the same orbit of (3) as the edge ofthe same length in Pt

8. R1 $ R2 contaim ezactly two edges of each of the lengths in LC, one from each

of the two orbits of (p2 )

5. among the four vertices of degree 1 in R1 $R2 ezactly two have odd subscripts.

PROOF. Conditions 1 - 3 imply that

is a partition of the edges of X(n - 2; LC) $ (p2) (P) .

Xow let the endpoints of the path R, be us, and ut,, and let the vertices of

be v and w .

By Condition 5, without loss of generality, only two cases may arise: either sl

and tl are odd and $2 and t2 are even, or sl and $2 are odd and t l and t z are even.

In both cases, if we define the cycle C by

it is not difficult to see that IpZi(C) : i = 0,. . . , -- *-' 2 1) is a C,-decomposition of

(x(n - 2; Lc) @ (p2)(p)) w G. 0

2.3.2 Central cycles

In the next two lemmas we describe the construction of central cycles for rt 2 3 and r' = 1, respectively. In the next section we then show that peripheral cycles can be

found so that the remaining edge lengths satisfy the conditions of Lemma 2.3.2 and

Lemma 2.3.5, respectively.

Lemma 2.3.2 Let r' 2 3. Define a zig-rag mt-path Poa by

Po,o = uo ul u 21 112 U- + .I 1 - *UZL=A U+l) K++D* 2

Furthermore, let LC be a set satisfying the following conditions:

I . ILcl = f ( m - 4 Om'),

2. LC = Lg U LA, when Lg = (2,. . . , ), and

m'-l 3. D - 2 ~ LA { T + $ ,..., D - 2 ) .

Figure 2.8: Lemma 2.3.9: the rig-zag m'-path PoVo.

PROOF. Fint observe that the edge length set of Po,o is

so that Lo n LC = 0. It thus makes sense to define the graph G = X ( n - 2; LC) $

(p2 ) (p-' (ha)). We shall describe the paths R1 and R2 that satisfy the conditions

of Lemma 2.3.1 for this graph.

The following notation will be used:

N = l ~ ~ l = t ( ~ - 4 - m ' ) - a = 2 f ( m -- m' - T ' - 3),

LA = {a1,. . . ,ON+ D - 21, where 9 + $ 5 a1 < . < q v - 1 < D - 2,

A2 = - $+a l -az+*e*+( - - l )NaN- l ,

A = A2 + ( - I ) ~ + ' ( D - 2),

B = + + k L z l = 5 + . . . 2 2 + (-1)*3 + (-1)*2, and

B* = ~ + ( - l ) + .

The numbers A, A2, B, and Bl, evaluated in the integers, lie on the interval (4, D). Thus, for example, A < 0 means the vertex u~ is in the set {uD+1, U D + ~ , . . . , u2D-1}

while A > 0 means the vertex u~ is in the set {ul, u2,. . . , uo-1).

Let PA2, PA, PB, and Q denote the following zig-zag paths:

These paths will be used to construct the paths R1 and Rz. To connect them, we shall

use edges of lengths 1 and D - 1 from the same orbits of (p2) as the corresponding

edges in p-l(Po.o); that is, from the orbits (p2)(u-luo) and (p2)(u-+u-++D-,) In some cases, two edges of length D - 2 from distinct orbits of ( p 2 ) will also be used

as linking edges.

For ease of reference, let us list the edge length sets of the zig-zag paths fA2, PA, PB, and Q:

L(PA1) = {a19 , U N - 1 1 , L(PA) = LA = {al, . . . ,ON+ D - 21,

L(Pe) = Lg = {2,3 ,..., 91, and m'-3 L(Q) = {T ,..., T+$}= ~ ~ - { l , ~ - l } .

The paths R1 and R2 will be constructed in such a way that Conditions 1 - 3 of Lemma 2.3.1 are satisfied for the edge length sets Lo and LC, and the m'-path

P = p-l(Po,o). Notice that this implies that the total number of edges in Rl $ R2 is

which satisfies Condition 4 of the same lemma.

The details of the construction of the paths R1 and R2 depend on the residue

class of n' modulo 4, and on whether A and N are odd or even (notice that N is odd

if and only if A > 0). Keep in mind that in order to be able to employ Lemma 2.3.1,

we also need to make sure that Condition 5 is satisfied; that is, that exactly two of

the four endpoints of the paths R1 and R? have odd subscripts.

We now describe the paths Rl and R2 for the various cases. In each case we

examine the subscripts of the endpoints and the orbits of the linking edges; that is,

the edges of lengths 1 and D - 1, and, in some cases, D - 2.

1. Case n' n O(mod 4). This implies $ and D are even.

1.1. A odd. Let

Since r' 2 3, it follows that m' 2 5 which in turn implies that R1 and R2 are indeed

vertex-disjoint . The subscripts of the endpoints are: B, B + D + 1 (one odd, one even), D + 1

(odd), and - $ + D + 2 (even).

We determine the orbits of ( p Z ) containing the !inking edges:

Although Figure 2.9 shows this construction only for N odd (that is, for A > O),

the above definition of Rl and Rz clearly works for N even as well.

1.2. A even. Let

Figure 2.9: R1 and Rz for Case 1.1: n' a O(mod 4), A odd (subcase N odd shown).

The subscripts of the endpoints are: B, B + D - 1 (one odd, one even), D - 1 (odd), and - $ + D (even).

W e verify the orbits of (p2) containing the linking edges:

Again, Figure 2.10 shows this construction only for N odd (that is, for A > O),

although it works just as well for N even.

Figure 2.10: R1 and Rz for Case 1.2: n' a O(mod 4), A euen (subcase N odd shown).

2. Case n' s 2(mod 4). This implies 5 and D are odd.

2.1. Subcase N even. Hence A c 0.

2.1.1. A odd. Let

Fi y r e 2.11: R1 and R2 for Cue 2.1.1: n' I 2(mod 4), N even, A odd.

The subscripts of the endpoints are: B, B + D (one odd, one even), D (odd), and - $ + D (even).

The orbits of (p2 ) containing the linking edges are:

(P*)(UAUA+D-I) = ( P ~ ) ( P ( u o u D - I ) ) = (p2)(u+ y-+ + D - I ) and

(p2) (u+D-IUA+D) = (p2) (u- I UO).

Figure 2.12: R1 and Rz for Case 2.1.9: n' I ?(mod 4), N even, A even.

2.1.2. .4 even. Let

The subscripts of the endpoints ate: B, B + D (one odd, one even), D (odd), and - f + D (even).

We determine the orbits of (p2) containing the linking edges:

Figure 2.13: R1 and R2 for Case 2.2.1: n' n 2(mod 4), N odd, A2 odd.

2.2. Subcase N odd. Hence A = A2 + D - 2 and A2 < 0.

2.2.1. Az odd. Let

Observe that since n' = 2(mod 4) and r' 2 3, rn' 2 7. Hence and pD-2(Pg) are vertex-disjoint.

The subscripts of the endpoints are: Bl, B1 + D - 2 (one odd, one even), D - 2 (odd), and - $ + D (even).

The orbits of ( d ) containing the linking edges are:

2.2.2. A2 even. Let

The subscripts of the endpoints are: Bl, Bl + D - 2 (one odd, one even), D - 2 (odd), and - 5 + D - 2 (even).

The orbits of ( p 2 ) containing the linking edges are:

We have thus constructed the paths Rl and 4 that satisfy the conditions of

Lemma 2.3.1. Therefore

Figure 2.14: Rl and R2 for Case 2.2.2: n' = ?(mod 4), N odd, A2 even.

Next we construct central cycles for the case r' = 1.

Lemma 2.3.3 Define o zig-zag m'-path by

and let LC be a set satisfying the following conditions:

2. LC {m', . . . , D - 21, and

9. LC {n', . . . , D - 2 ) for m' = 3.

PROOF. Notice that PoVo of Lemma 2.3.2 for r' = 1 yields precisely the same m'-path

as defined here. Its edge length set is

Lo= {1, ..., ml- 1 ,D-1)

so that Lo n LC = 0 and it makes sense to define the graph G = X(n - 2: LC) $

( P ~ ) ( P - ' (~04))- As in the proof of Lemma 2.3.2 we shall construct the paths R1 and R2 that

satisfy the conditions of Lemma 2.3.1 for the edge length sets Lo and LC, and the

zig-zag m'-path P = p-'(Po,o), but first we introduce the notation:

N = lLcl = i ( m - 4 - m ' ) ,

LC = {al,. . . , aN}, where m' 5 a1 < . . . < 5 D - 2, and

A = -$ + a1 - a* + + ( - 1 ) ~ + ' 4 ~ .

Let PA and Q denote the following zig-zag paths:

Since n' = m' + 1, we can see that p-l(Por) = Qu-+u-+ +D-l. The edge used to

link PA and Q into R1 and 4 will come from the orbit of (P?) containing the edge

u+u-$+D-l As in Lemma 2.3.2, the paths R1 and R2 will be constructed in such a way that

Conditions 1 - 3 of Lemma 2.3.1 are satisfied. Since

this implies I E(RI $ Rz) 1 = m - 4 so that Condition 4 is satisfied as well. The subscripts of the endpoints of Rl and Rz will be verified within each case to see that

Condition 5 is met, too.

1. Case 11' 1 O(mod 4). This implies $ and D are even. Let

Rl = u ~ u - ~ Q P ~ and

R2 = p D - l ( ~ A ) .

Since a1 1 rnt = n' - 1 for mt 2 5 and a1 2 nt for m' = 3 by conditions 2 and n ' 3, -- 2 + a ~ 2 2. Hence -5 + a~ + D - 1 2 D + 1 > D so that R1 and Rz are

vertex-disjoint . The subscripts of the endpoints are: A, A + D - 1 (one odd, one even), D (even),

and -5 + D - 1 (odd).

We now determine the orbit of ( p 2 ) containing the linking edge:

vaticu with

b oddsubmipa

Figure 2.16: R1 and R2 for Case 2: n' E 2(mod 4) (subcase A > 0 shown here).

We have thus constructed the paths R1 and R2 that satisfy the conditions of

Lemma 2.3.1. Therefore

2.3.3 Peripheral cycles

In the next lemma we show how to construct peripheral cycles for both r' 2 3 and r' = I.

Lemma 2.3.4 As in Lemma 2.3.2, let the zag-zag m'-path Po,0 be

and denote its edge length set b y Lo. There ezists a set L p with the following

properties:

9. I LPI = Fm', and

PROOF. First we construct the solitary peripheral cycles. Observe that the edge

length set of POSO as defined above is

Now let Potj = @n' (~Op) for j = 1,. . . , d - 1. Since the longest edge in Pea, not counting the edge of length D - 1, has length

the paths Poj are pairwise vertex-disjoint except for the endsoints. Since gcd(d, y) = 1, Co = $::A Poj is an rn-cycle. Hence

Next we describe the coupled peripheral cycles. First assume that rn' 2 5 and t' 2 5. Define the zigzag (m' - 1)-path Pip by

Figure 2.17: Lemma 23.4: the zig-zag (m' - 1)-path for m' 2 5, t' 2 5 , rn' 1 (mod 4).

if m' r 1 (mod 4), and by

if rn' r 3 (mod 4). The edge length set of the path is

L1= { 2 + n', 4 + n',5 + n', . ,m' + 1 + n') if m' ZE 1 (mod 4), and

if m' E 3 (mod 4).

For j = I , . . . , d - l let Plj = p'"(fi,o). Sincer' >_ 5, n ' - F > 2 1 implying that in both cases these paths are pairwise vertex-disjoint .

For i = 2,. . . , C-' 2 9 obtain the zig-zag (m' - 1)-path Piso from PI,* by adding ( i - 1)n' to the subscripts of the even vertices. The paths in each of the families

Pi = { ~ ' ( f i , ~ ) : j = 0,. . . , d - 1) an thus pairwise vertex-disjoint. The edge length

set of PiPo is Li = L1 + ( i - 1)n'. We shall use the technique of Paragraph 1.4.1.1 to connect each family of (m' - 1)-

paths into an m-cycle. If 9 is even, use the auxiliary circulant X = X(d; {2,3, . . . , F}) and decompose it into circulants X(d; {2,3)), . . . , X(d; (9, y}), or X ( 4 {2)),

Figure 2.18: Lemma 2.9.4: the rig-zag (mt - 1)-path PIqo for m' > 5, rt 2 5 , m' 1 3 (mod 4).

c-l fi X(d; {3,4}), . . . , X(d; { T , I) , depending on whether 9 is even or odd. Since

each of these circulants of degree 4 is connected and hence decomposable into Hamil-

ton cycles by Theorem 1.1.2, we have a Hamilton decomposition of X. Since the

terminal vertex of each of the paths Rqo is u-1, an edge of length 1 in a d-cycle C in this decomposition gives rise to a connecting edge of length in' - 1 or in' + 1,

depending on the orientation of C we are using as explained in Paragraph 1.4.1 .I.

in this case, use each d-cycle in the Hamilton decomposition of X with both orien-

tations, thus connecting two families of (mt - 1)-paths. The set of lengths of the

connecting edges is thus

If 9 is odd, use the auxiliary circulant X = X(d; {1,2,3, . . . , }) asd decom-

pose it into X(d; {I}), X(d; {2,3)), . . . , X(d; (9, y}), or X(d ; {I)), X(d; {2)),

X(d; {3,4)), . . . , X(d; {F, )), depending on whether 9 is odd or even. Obtain a Hamilton decomposition of X as before and then use each d-cycle in this decom-

position with both orientations except for the d-cycle X(d; {I)), which we use only

with the orientation resulting in connecting edges of length n' + 1. The set of lengths

of the connecting edges is thus

Lx = {n' + 1) U {f 1 +in f : i = 2,.. ., 9).

c-a

Since r' 2 5, the sets Lx and uif; Li are disjoint in all cases. Hence the set

Lp = (U,z L i ) U Lx has size 9 m f , satisfying Condition 2 of the lemma. Let Ci be the m-cycle arising from the family Pi of (m' - 1)-paths.Then

is a C,-decomposition of the circulant X(n - 2; L p ) . Since the shortest edge of

X(n - 2; L p ) has length at least n' + 1 and the longest edge has length at most

Lp satisfies Condition 1 of the lemma. Since Lp and Lo are disjoint, it makes sense

to define the graph X(n - 2; Lp) $ (p2)(Pop) and this graph is C,-decomposable as

shown above. Hence the set L p satisfies all conditions.

Figure 2.19: Lemma 2.3.4: the zipzag (m' - 1)-path PI,* for r' 5 3, m' 2 5.

Next, let m' > 5 with r' 5 3 . Define the zig-zag (m' - 1)-path PIp by

Its edge length set is

Forj = I , ..., d-l let P I j =p'"'(qqo). ~ i n c e ~ + 2 n ' + ~ = m ' - 1 + 2 n f < 3 ~ ~ , these paths are pairwise vertex-disjoint.

For i = 2,. . . , 9, obtain the zig-zag (m' - 1)-path from qp by adding

2(i - 1)n' to the subscripts of the even vertices. The paths in each of the families

Pi = {p'"(&) : j = 0,. . . , d - 1) are thus pairwise vertex-disjoint and the edge

length set of Pi,o is Li = L1 + 2(2 - 1)n'.

To connect the family Pi of (m' - 1)-paths into an m-cycle we use the same

auxiliary circulant with the same Hamilton decomposition as before. The only

difference is that now the lengths of the connecting edges arising from an edge of

length I in X will be ( I - 1 )nt - 1 and ( I + 1)n' + 1 since the terminal vertex of the

path Pi,* is u.l+l. The d-cycle X(d; {I ) ) should now be taken with the orientation

that results in connecting edges of length 272' + 1. We thus have

if is even, and

L~ = {-I + int : i = 1,. . . , "}u{l+ 4 2 n t : i = 2 , ...,

if 9 is odd. e-I c-1

Again, the sets Lx and U,z L; are disjoint so that L p = (U,z L i ) U LX is of size

a m ' 2 and X(n - 2; L p ) is C,-decomposable. The shortest edge of X(n - 2; L p ) now r'd 2 has length n' - 1 while the longest edge has length m'- 1 + ( c - l )n t . Since c = *,

we can show that c < f in all cases except for m' = 5, r' = 3 as follows. If r' = 1,

clearly c = 9 < < f . If r' = 3 and rn' 2 9, c = m1 < - 9 < f . If rr = 3

and m' = 7, c = < f since d 2 11. Hence the longest edge in X(n - 2; L p ) has

length

This now implies that Conditions 1 - 4 are satisfied in all cues except for m' = 5, r' = 3.

Its edge length set is

Obtain PiVo from PIVo by adding ( i - I)n' to the subscripts of the even vertices so t

the edge length set of is L; = Ll +(i- l)nl, and then obtain Pi,j, j = 1,. . . , d- 1,

in the usual way. To link the paths Pi j into an m-cycle, if we now use precisely the c-1

same method as in the case r' 2 5, the sets Lx and u,Z L; will be disjoint. Hence c- 1

X(n - 2; L p ) , where L p = (Utz L i ) U L x , is C,-decomposable. The shortest edge

length in X(n - 2; L p ) is at least n' + 1, while the longest is

It now follows that Conditions 1 - 4 are satisfied.

Finally, let m' = 3 and hence n' = 4. Let the zigzag 2-paths and PCo be

Their edge length sets are

L1 = {2 + 2nt7 - 1 + 3n') and

L; = {I + 3nt,2 + 372'). For i = 2, . . . , obtain Pi,* from Pla by adding 2( i - l )n' to the subscript of the

second vertex, and for i = 2,. . . , LyJ obtain P:o from Pi,* in the same way. Then let Pi = {pi"'(pic) : j = 0,. . . , d - 1) and F: = {p'"'(~:~) : j = 0, . . . , d - 1).

We now use the auxiliary circulant X = X(d;{1,2,4,. . . ,2LyJ}) to choose

the linking edges. If 9 is even, decompose X into X(d; {1,2)), X(d; {4,6)), c-7 c-5 . . . , X(d; (9, 9 )), or X(d; {I)), X(d; {2,4)), . . . , X(d; {T, depending on

whether 9 is odd or even. Using every d-cycle in the Hamilton decomposition of X thus obtained with both orientations results in the set of lengths of the connecting

edges being

c-5 c-3 If 9 is odd, decompose X into X(d; ( I ) ) , X(d; {2,4)), . . . , X ( d ; iT , or e-5 c-3 X(d; {I)), X(d; {2)), X(d; {4,6)), . . . , X(d; { T , depending on whether 9 is

even or odd. Take the d-cycle X(d; (1)) with the orientation resulting in connecting

edges of length nt - 1 and all other cycles in the Hamilton decomposition of X with

both orientations. The set of lengths of the connecting edges is thus

L3) are disjoint. Let Observe that in both cases Lx and (UiZ1 Li ) U (Uisl L p be their union. The set Lp thus satisfies Conditions 2 and 3 of the lemma and

X(n - 2; L p ) is C,-decomposable.

The smallest element of L p is clearly n' - 1, while the largest is at most

since c = and d 2 7. Hence Conditions 1 and 4 are satisfied as well.

This proves the lemma for all cases.

2.3.4 Conclusion

Finally, we show that the collection of centrd cycles in Lemma 2.3.2 or Lemma 2.3.3

and the collection of peripheral cycles in Lemma 2.3.4 form a C,-decomposition of

K , - I .

Lemma 2.3.5 Let n be an even integer and m be an odd integer such that 3 5 m 5 n < 2m and n(n - 2) 0 (mod 2 m ) . Then K, - I b C,-decomposable.

PROOF. Define the parameters r, d, m', n', r', c, and D as in Section 2.3.1. As we

have seen in the discussion on page 36, d 1 3 and m' 2 3.

Observe that K,-I is isomorphic to X(n-2; L) w K*, where L = {I,. . . , y-1). Let the zig-zag m'-path Sp, its edge length set Lo, and the edge length set Lp be

as in Lemma 2.3.4. Let LC = L - ( L p u L O ) . It is not difficult to see that Conditions

2 and 3 of Lemmas 2.3.2 and 2.3.3 are satisfied for r' 2 3 and rt = 1, respectively.

Since 1 Lo 1 = mt and 1 Lp ( = Fm',

satisfying Condition L of Lemmas 2.3.2 and 2.3.3 as well. The graph X(n - 2; L) w K~ can now be partitioned into

which is C,-decomposable by Lemmas 2.3.2 and 2.3.3, and

which is C,-decomposable by Lemma 2.3.4. Hence K,, - I is C,-decomposable.

0

Chapter 3

Decomposit.ion of K, into m-cycles, where n is odd and m is even

3.1 Inductive step and the main result

Fi yre 3.1: From a decomposition of K, to a decomposition of Kn+h .

Unlike in the problem of a C,-decomposition of K, - I for rn odd, n even, the

induction step for the C,-decomposition of K,, m even, n odd, is easy and has

already been demonstrated by C. A. Rodger in [20]. We present it in a slightly

different form.

Theorem 3.1.1 [20] Let n be an odd integer and m be an even integer such that 3 5 m 5 n. If K. is C,-decomposable then so is Kn+2,.

PROOF. It is not difficult to see that the edge set of the complete gaph Kn+2rn cam

be partitioned into three sets: the edge set of the complete graph K,, the edge set

of the complete gaph K2m+l (where these two have a vertex in common), and the

edge set of the complete bipartite graph Kn-1,2m (Figure 3.1).

Kn is C,-decomposable by assumption.

Figure 3.2: A C,-decomposition of K2,+1 for m 0 (mod 4) and m 1 2 (mod 4).

K2m+I is well known to be C,-decomposable [15,21]; one such decomposition is

: i = 0,. . . , n - 11, where

if m e 0 (mod 4), and

C = uo ul u-1 u u 6 ... U f - 1 u-f U y + 1 U*

if m 1 2 (mod 4). (Figure 3.2)

Finally, since m is even, n - 1 2 7, 2m 2 ?, and 2m(n - 1) r 0 (mod m ) ,

Kn-l,lm is C,-decomposable by a well-known theorem by Sotteau [24].

Hence &+lm is Cm-decomposable. 0

The above theorem tells us that it is sufficient to find C,-decompositions of

K, for all n in the range m 5 n < 3m that satisfy the necessary condition n(n - 1) E 0 (mod 2m) and this will be our preoccupation for the rest of the chapter.

The following theorem, which is the main result of this chapter, then draws the

conclusion.

Theorem 3.1.2 Let n be an odd integer and m be an even integer such that 3 5 m 5 n and n(n - 1 ) = 0 (mod 2m). Then K,, is C,-decomposable.

PROOF. Assume that n 1 rn satisfies the necessary condition n(n - 1) 0 (mod 2m)

and write n = q - 2m + p, where m 5 p < 3m. Since n(n - 1) m p ( p - 1) (mod 2m),

p satisfies the necessary condition p(p - 1) = 0 (mod 2m) as well. Hence. if m 5 p < 2m, K p is Cm-decomposable by Lemma 3.3.15, and if 2m 5 p < 3m, & is C,-decomposable by Lemma 3.2.4. By successively applying Theorem 3.1.1 q times

we can now establish that K, = Kg.*,+, is C,-decomposable. 0

3.2 C,-decomposition of Kn for n odd, m even, and 2m 5 n < 3m

3.2.1 Central cycles

The construction for the case 2m 5 n < 3m involves two types of cycles: central

cycles, described in the next lemma, and peripheral cycles, which take a lot more

work and are described in the next two sections.

For an example of a central cycle see Figure 3.9.

Lemma 3.2.1 Let LC = {al,az,. . . ,am), when a1 < a2 c . . . < a,, be a subset of the edge length set (1, 2, . . . , *) 2 and denote Si = a, - a2 + + (-l)'+'ai for

i = 1, . . . , rn. Let the following conditions be satisfied:

2. there erists an even integer k such that ai+l - ai = 1 for all i = k + 1, k + 2, ..., rn- 1, and

Then X(n; LC) is C,-decomposable.

PROOF. Let 6 be either 0 or 1, whichever makes S, - 6 even, and let S = (S, - 6).

For any j E (1, . . . , m - 11, let Rj+l and x ( j ) denote the following:

Whenever j is odd and k + 1 5 j 5 m - 1 we have .

We want to find an odd integer j, k + 1 < j 2 m - 1, such that

that is,

Since

Sj-1 = Sk - z and m-k Rj+l r -aj - - -

2 6 + + ,

there exists an odd integer j, k + 1 5 j 5 m - 1, satisfying (3.3) if and only if there

exists an integer 2, 0 5 x f (m - k - 2), such that

The solution to this equation is

which is in the appropriate range if and only if Sk + 9 + 6 > 0 (observe that Sk is negative). This condition is equivalent to ab+1 - 2Sk - 26 5 since m - I; = n-1 m k - (ak+l - 1). Furthermore, Sk + + + 6 is even by (3.1) so that z is an integer.

Since by Condition (3) of the lemma, ak+l- 2Sk _< y, we thus have an odd integer

j , k + 15 j 5 m - 1, satisfying 3.2.

This means that

is an m-cycle. Since the length of the edge

is 9 for both 6 = 0 and 6 = 1, the edge length set of C is precisely LC =

{ a l , a 2 , . . . ,a,]. Hence

{$(c) : i = 0,*.. ,?2 - 1)

In the next two sections we construct peripheral cycles with the property that the

remaining edge lengths in K, satisfy the conditions of Lemma 3.2.1 and therefore

can be used to construct central cycles. We shall prove this by finding the even

integer k of Condition 2 and showing that ak+~ - 2Sk y. Before we move on, however, let us have a quick look at the parameters involved.

Let r = n - 2m. Notice that r is odd, and since the case r = 1 has been solved (see

the proof of Theorem 3.1.1), we have 3 5 r < m. Let d = gcd(m,n). Now d = 1

and 2mln(n - 1) imply 2m((n - 1); that is, n = 2m + 1, so we may assume that d,

which is odd, is at least 3.

Notice that d = gcd(m,r) as well. Let n = n'd, m = m'd, and r = r'd. Then 2mln(n - 1 ) implies 2mlr(r - I), whence 2m'l(r - 1). Hence there exists a positive

integer c such that r - 1 = 2m'. Since r < m, we have 2m' = r - 1 < m'd - 1,

whence c 5 y. The construction of the peripheral cycles now splits into two cases: c 5 9 and d-1 c > T .

3.2.2 Peripheral cycles for the case c 5

The following inequalities will be used throughout this section to prove that ak+1 - 2Sc 5 for the different constructions and parameter values:

Lemma 3.2.2 Let n be en odd integer and m be an even integer such that 2m 5 n < 3m and n(n - 1 ) r 0 (mod 2m), and let c be as defined in Section 9.2.1. If c 5 9, then h:, is C,-decomposable.

Figure 3.3: Lemma 9.2.2: the rig-zag m'-path Poo for m' n 2 (mod 4), m' > 2.

PROOF. We shall prove the lemma by constructing peripheral cycles for the various

cases. In each case we show that the remaining edge lengths satisfy the conditions

of Lemma 3.2.1 and hence can be used up in central cycles.

CASE 1. rn' a 2 (mod 4), rn' > 2. Let

3 t 3 t Since n' - (5 + 1) = ~ r n + r' - 1 2 pa > rnt + 1, PoVo uses precisely mf distinct

edge lengths. Furthermore, since the length of the edge u-+ u ++, is m' + 1 < n', the sets of the internal vertices of the paths Poj = @"'(P&O), j = 0, . . . , d - 1, are pairwise disjoint. Hence Co = $$A Poj is an m-cycle.

For i = I,. . . , c - 1 obtain Pivo from Po,* by adding in' to the subscripts of the

even vertices. By the observation above, Ci = $::: R j is an m-cycle for every i.

Let L, denote the edge length set of the zig-zag m'-path Pi,, We have

and, for i = 1,. . . , c - 1,

C Clearly, these sets are pairwise disjoint. Hence {p'(C,) : i = 0,. . . , c - 1, j - 0, . . . , n' - 1) is a C,-decomposition of X(n; L p ) , where L p = u::; Li.

We proceed to show that the set LC = 11,. . . ,?) - L p satisfies the conditions

of Lemma 3.2.1 and hence that X(n; LC) is C,-decomposable as well. Clearly ILc I = n-l -- m' = n-' a

2 2 2 = m. Let LC = {al, . . . , a,), where a1 < . . . < a,. Notice d l t that max(Lp) = m' + 1 + (c - l)nt < cnt 5 7-n < 9 so that Condition (1) of

Lemma 3.2.1 is satisfied. We shall find the alternating sum Sk and the element ak+l

for the various cases to verify Conditions (2) and (3) of the same lemma.

For i = 0,. . . ,c - 2 let Ai denote the alternating sum of the edge lengths in

LC n (in', ( i + l)nl] and let denote the alternating s u m of the edge lengths in

LC n ((c - l)nf, max(Lp)]. For c 1 2 we have

= (2 + in') - (rn' + 2 + in') + + (n' + in') = f n' - tm' + 2 4- in'

f o r i = 1, ..., c-2 , and

A,-* = ( 2 + (c- l)nt) - ($ + 1 + ( c - 1)n') = - ) m r + 1.

Notice that the index k from Lemma 3.2.1 has to be even so that Sk and ak+l

depend on whether c is odd or even.

Subcase c even. We have .

and

uk+l = rnr + 2 + (c - 1)n'. Hence, by (3.5),

Subcase c odd, c 2 3. Since k has to be even, we take

Sk = A. + A1 - A2 + + As-2 - - (rn' + 2 + (c - l)n') and

Then

and, using (3 .5) and mt 1 6 ,

Subcase c = 1. Now

and

Hence

Figure 3.4: Lemma 9.2.2: the zig-zag mt-path Po,o for m' = 0 (mod 4).

CASE 2 . rnt I O (mod 4). We now let

Po.0 = uo u-1. . . u-+up+, U_(+_,, u ?+I 1 U*'

and create the zig-zag m'-paths P;: and the m-cycles Ci = $:s Pij as before.

The edge length sets are now

and, for i = 1,. . . , c - 1,

m' I m' Li = (1 + in'. 2 + in', . . . ,T + in' 2 + 1 + ( i - l)n ,T + 2 + in', . . . , m' + in').

Again we let L p = U:;; Li and conclude that X(n; L p ) is Cm-decomposable. We

then find the alternating sums Ai (for i = 0, . . . , c - 1), Sk and ar+l to show that

ak+l - 2Sk < y, thus proving that the set LC = (1, . . . , e) 2 - L p satisfies the

conditions of Lemma 3.2.1 and hence that X(n; LC) is C,-decomposable as well.

For c 2 2 we have

Ai = (m' + 1 + int) - - + (n' + in') = fn' + +m' + + + ;n'

for i = 1,. .. , c - 2, and

Subcase c even. We have

Sk = A. + A1 - A2 + - A,-* + - (m' + 1 + (c - l)nt) =

= ( - in t + 3m'- a) - a n ' 2 + (id+ 1 + (c- l)nt) - (mt+ 1 + (c- 1)n')

= -c-ln'- 1 2 2

and

a)+1 = m' + 2 + (c - l)nt.

Hence, by (3.5),

Subcase c odd, c 2 3. Now

and

ar+l = rn' + 1 + (c - l ) n t .

We obtain

so that by (3.6)

a k + 1 - 2& = 2(c - l ) n f + 3 < 2cnt < 9.

Subcase c = 1. In this case we have

m ' l t 1 3 Sk = ( $ + l ) - ( m ' + l ) + - - ( 1 2 ' - 3 - - 2 ) ' = -?n +;n'+ 5

and 1 r ak+1 = n' - ~ r n ,

whence

a)+, - 2Sk = 2n' - rn' - 3 < 2cn' < 9 by (3.6).

CASE 3. rn' = 2. This case, as tends to happen for small values of the parameter

m', requires a different construct ion. First observe that r' = 1 and therefore c = y. Furthermore, n' = 5. Let

and for i 2 1,

Note that the PiVo will be used for i = 0 , . . . , [$l - 1, whereas F':o will be used only

for i = 0,. . . , Lfj - 1. The paths Pi,. and cj, and the m-cycles C; = $$; Pi j and Cr = @% P:j u e created as before.

The edge length sets of the zigzag paths PI:,* and PCo are

and for i 2 1,

Li = (1 + lOi,6 + 10(i - 1) ) and

L = {2 + lOi, 7 + l O ( 2 - 1)) .

Ifl-1 We let L p = (UimO L~)U(U:!~-' LT) and conclude that X(n; Lp ) is C,-decomposable.

We now verify the conditions of Lemma 3.2.1 for the set LC = 11,. . . , 9) - L p . First we find the alternating sums A;. In this case it is convenient to let each A,?

i 2 1, pick up the remaining edge lengths on the interval [l+n'+2(i-l)nt, nt+2in'] =

(6 + 10(i - l ) , 5 + 10i] rather than the usual [l + in', n' + in']. For c > 2 we thus have

A. = 5 ,

Ai = (8 + 10(i - 1 ) ) - (9 + 10(i - 1 ) ) + (102) - (3 + 102) + (4 + 10i) - ( 5 + 1Oi) = -5

for i = 1,. . . , -2,

if c is odd, and

if c is even.

Subcase c even, c > 2. We have

and

whence

Subcave c = 2. Now St = 0 and ak+l = 5 , whence

ak+1 -2sk = 5 = h'< y.

Subcaae c odd, c > 1. In this case

and

whence by (3.5)

Subcase c = 1. Now Sk = 2 - 3 = -1 and ak+l = 5 , therefore ar+l - 2Sk = 7 < 2cn' < y.

We have shown that in all cases X(n; L p ) is C,-decomposable and that the set

of the remaining edge lengths LC = {I,. . . , d) 2 - L p satisfies the conditions of #

Lemma 3.2.1 so that X(n; LC) is C,-decomposable as well.

Hence h;, is C,-decomposable. 0

3.2.3 Peripheral cycles for the case c >

Notice that in the constructions of peripheral cycles for the cases with c 9 (except for m' = 2) , each zig-zag path Pito, with the exception of at most one edge,

used only edge lengths from the set {I +int, . . . , n' +int) . For the cases with c > 9, however, we will have two zig-zag paths Pito and co corresponding to one section

of n' vertices.

Recall that c 5 y. The following inequalities will be useful in proving that

ak+1 - 2sk 5 9 (that is, that the edges remaining after the construction of pe-

ripheral cycles can be partitioned into central cycles):

The latter follows from r'd - 1 = 2 m ' 2 2 Y m t .

Lemma 3.2.3 Let n be an odd integer and rn be an even integer such that 6 < 2m 5 n < 3m and n(n - 1 ) I 0 (mod 2m), and let c be as defined in Section 3.2 1.

If c > v, then h;, is C, -decomposable.

PROOF. As in Lemma 3.2.2, we shdl prove the statement by constructing periph-

eral cycles and showing that the remaining edge lengths satisfy the conditions of

Lemma 3.2.1. d-1 CASE 1. m' P 2 (mod 4). Since m' = 2 implies t' = 1 and hence c = T , we

may assume that m' 2 6. Furthermore, since r' 2 $ + 1 by (3.8) and r' is odd,

r' 2 $ + 2 throughout this case. In particular, r' 2 5 and n' 1 17.

Denote t = n' - m' - 3 and let

Obtain Pl,, from Po,o by adding t to the subscripts of the even vertices. That is, let

Figure 3.5: Lemma 9.2.9: the rig-rag m'-paths Pas and P&, for m' a 2 (mod 4), c > 1.

Since n' - ($ + 2) > rn' + 2, Poa uses precisely rn' distinct edge lengths. Since

n' - ($ + 2 + t ) = $ + 1 < n' - mt - 2 = 1 + t, P;lo uses precisely rn' distinct edge

lengths. As usual, we let Poj = @"'(P~,~) and Pitj = p'"'(~t~,). Since m' + 2 < n', the sets of internd vertices of the paths Poj are pairwise disjoint. Similarly, since

m' + 2 + t = n' - 1 < n', the sets of internal vertices of the paths P;,j are pairwise

disjoint. Hence Co = @, Po,. and C; = @;:; P;,, are m-cycles.

Notice that n' - ($ + 2) = % + 1 + t and n' - (* + 2 + t ) = $ + 1. The following are the edge length sets of the zigzag d p a t b s and P;,j:

and m ' LC= { $ + l , l + t , 3 + t ,..., T + t , $ + 3 + t ,..., m t + 2 + t ) .

Since r' 2 5, m' + 2 < 1 + t and these two sets are disjoint.

For i = 1,. . . , - 2, we obtain P , , O horn Polo and P:o from P&, by adding in'

to the subscripts of the even vertices. By the observation above, Ci = $::A Rj and C; = $::; Pri are rn-cycles.

The edge length sets of the zig-zag m'-paths Piqo and PI:, are

Li = {1+int ,3+in' ,..., $ + i n ' , $ + 2 + ( i - l ) n t ,

+ 3 + in', . . . , m' + 2 + int) 2

and

Finally, let

and obtain Pkl from Pr tl -l,o by adding t to the subscripts of the even vertices.

These two zig-zag mt-paths then give rise to the m-cycles Crtl-l,o and Ci51-,,o in

the usual fashion. Note that Ciil-, will be used in the construction only for c even.

The edge length sets of qtl -1 ,o and Pit1 are

and

Since the Li and Lr are pairwise disjoint for all i,

r 31-1 tt~-1 L. is a C,-d~omposition of X(n; L p ) , where Lp = (UiZo Li) U (Ui=o i ) *

To see that X(n; LC), where LC = {I , . . . , -) 2 - L p , ig C,-decomposable, we

again use Lemma 3.2.1. First we shad find the alternating s u m s Ai7 which pick up

the remaining edge lengths on the intervals [1+ in', n' + in'].

For c 2 4 we have

Ai = (2 + in') - ($ + 1 + in') + ( m r + 3 + in') - - -(t + i n t ) + (2 + t + int) - ($ + 1 + t +in') + (n'+ in')

= l n r + g + i n t 2

for i = 1, ..., Ljj - 2,

c-3 I ALfl-l = A=-3 = ( 2 + yn') - ($+ 1 + y n r ) + ( m t + 3 + ~ r t ) - . e W

T c-3 I c-3 t c-3 r -(t+Tn)+(2+t+Tn)-($+l+t+Tn)

c-3 r +($ + 2 + t + yn') - (n' + ~n )

- 7 - -in'- fm'+?

and c l c-l Al+l-l = Ac-I = ( 1 + g n ) - (2 + ~n ) = -1

-T

if c is odd, and

if c is even.

Since the integer k in Lemma 3.2.1 has to be even, Sk and ak+l will depend on

the residue class of c modulo 4.

Subcase c r 0 (mod 4). We have

whence

by (3.7).

Subcase c = 1 (mod 4), c 2 5. Now

and c-1 I ak+1 = rn' + 3 + ~n .

Since m' 2 6 and r' 2 5 , and by ( 3 3 ,

Subcase c 1 2 (mod 4), c 2 6. We have

thus, using (3.7) and nf 2 17,

Subcase c = 2. Using Co and C,' arising from the paths in (3.9) and (3.10) we obtain

and a k + l = n'. Hence

Subcase c r 3 (mod 4), c 2 7. Now

and

a k + ~ = m' + 2 + y n t .

Since m' > 6, by (3.7),

Subcase c = 3. Now

s k = 2 - ( m t + 3 ) + - m + t - ( 2 + t ) + ( % + 2 + t ) - n t + ( l + n t ) - ( 2 + n t )

- - --1 1 1 - 1 r - 3 2 zrn 2

and ak+t+l = mt + 2 + n'. Since r' 2 5,

Subcase c = 1. Using the zig-zag path Pogo as defined in (3.9) does not yield the desired result. Instead, we are going to use Prrl-ln as defined in (3.11) for c = 1,

namely,

= uo u3 u-1. . . U U*+2 un'

Figure 3.6: Lemma 9.2.9: the rig-zag ml-path PoVo for rn' i 2 (mod 4), c = 1 .

so that

We obtain

Sk = 1 - 2 + ( m t + ? ) - - - - (n' - $ - 3) 3 1 = -hz'+im 2 + I

with a ~ + ~ = n' - EL - 1. Since r' < $, 2

CASE 2. rn' r 0 (mod 4), m' 1 8. Recall from (3.8) that r' 1 $ + 1. We

thus have r' 2 5 and n' 2 21. This construction is very similar to the previous one.

Again, let t = n' -- m1 -- 3, but this time let

As before, P;,, is obtained from Po.o by adding t to the subscripts of the even vertices

so that

p<O = U~ '-1 ' U-* u$+3+1 * U - ( ~ - l ) U$+2+L

Figure 3.7: Lemma 3.2.3: Me zipzag mi-paths PoVo and Pi,* for m' = 0 (mod 4), m ' r 8.

Since n' - ( $ + 2) > m' + 1 , Povo uses precisely rn' distinct edge lengths. Since

n' - ($ + 2 + t ) = % + 1 < n' - rn' - 2 = 1 + t , P;, uses precisely m' distinct

edge lengths. Since m' + 1 < n', the sets of internal vertices of the P o , are pairwise

disjoint. Similarly, since m' + 1 + t = n' - 2 < n', the sets of internal vertices of the

P;,, are pairwise disjoint. Hence Co = @::; Poj and C,' = $:;; P,,, are rn-cycles.

~ ~ a i n , keepin mind that n t - ( $ + 2 ) = + + l + t and n ' - ( % + 2 + t ) = $+I .

The following are the edge length sets of PaVo and P;:,:

Since r' > 5, m' + 1 < 1 + t so that Lo and Li are disjoint.

For i = 1, . . . , ri] - 2, we obtain P;:,o from Pop, and yo from Pl,,, by adding

in' to the subscripts of the even vertices as in the previous case. By the observation

above, Ci = $;:A Pi j and C: = $;, qj are m-cycles.

The edge length sets of the zig-zag m'-paths PiVo and P& are

Li = { 1 + i n ' , 2 + i n t , ...,$+ i n t , + + 2 + ( i - l ) d ,

$ + 3 + in', . . . ,m'+ 1 + in') and

LT = { l + t + i n ' , 2 + t + i n ' , ...,%+ t + i n t , $ + 2 + t + ( i - l ) n ' ,

d + 3 + t + i n t , 2 ..., m'+1 + t + i n ' ) .

Finally, let

and obtain Pifl-l,o from Prgl-l,o by adding t to the subscripts of the even vertices.

The edge length sets of these zig-zag m'-paths are

and

Define the m-cycles CrP -1 and CfIl-l as before. Again, Cifl-l will be used in the

construction only for c even.

For c 2 5 , the sets L, and L? are pairwise disjoint. We thus have a Cm- r +1 LZ J - ~ LI) decomposition of X(n; L p ) , where L p = (Uiio Li) u (Uiz0

To see that X(n; LC), where LC = {I,. . . ,9=L} 2 - Lp, is C,-decomposable, we

proceed as in the previous case. First we find the alternating sums Ai. For c 1 5

we have

A. = (m' + 2) - + t - (n' - 1) + n' = in' + i.

Ai = ( ? + l + i n ' ) - ( m ' + 2 + i n ' ) + * * *

- ( t + in') + ($ + 1 + t + in') - (n' -- 1 + in') + (n' + in')

= 1,' + f + in' 2

for i = 1,. . . , - 3;

and c-1 t A r c = A c - I = 1 + ~ n

51 - -T

if c is odd; and

= ( $ + 2 + ( $ - 2 ) n t ) - ( m ' + 2 + ( 5 - 2 ) n t ) + . - - ( t + ( f - 2)nt)

+($ + ? + t + (5 -2)n') - (n' - 1 + ( f - 2)n') + ( a t + ( 5 - 2)nt) - c-3 1 5 - ~n + 5

and

if c is even.

Subcase c i 0 (mod 4), c 2 8. We have

Sk = A. - A1 + A2 - - A p 3 + - A p l - (n' - 2 + (5 - l ) n t )

= (f ++)- ; (n t+1 + ( t - 2 ) n t ) + ( 9 n ' + f ) - (-$+f) - ( -2+ fn')

= -cn'+3 4

and a&+, = fn' - 1,

whence, by (3.5), a&+l- 2Sk = cn' - 7 < 9.

Subcase c r 1 (mod 4), c 2 5. Now

and

Hence

Subcase c 2 (mod 4), c 2 6. We have

and

ak+1 = in' - 2,

thus by (3.7)

a k + ~ - 2sk =a'- 1 < y.

Subcase c r 3 (mod 4), c 2 7. Now

and c l t

ak+1 = m t + 2 + + n .

Since m' 2 8,

Subcase c = 1. We have

with ak+1 = $ + 3 + t = n1- Since d 2 3 and r' < 9, 2 '

Subcase c = 2. We obtain

and = n'. Hence

Subcase c = 3. We now use PI.p arising from (3.12) for i = 0, 1, and Pi,,:

Sk = ( m t + 2 ) - - + t - ( + + 2 + t ) + ( n t - 1 ) - n '

+ ( $ + l + n f ) - ( $ + 2 + n t )

= - ln'+fm'- E 2 2

and awl = m' + 2 + n'. Since n' 1 21,

Subcase c = 4. We use RPo and P& arising from (3.12) for i = 0,1. We obtain

sk = ~ ~ - ( $ + I + n ' ) + ( $ + 2 + n ' )

-(ml+ 2 + n') +- - ( t + n') + ($ + + + t + nl) - (% + 2 + t + n') = -nl+ 1

and ak+1 = m' + 2 + t + n' = 2n1 - 1. Hence

Figure 3.8: Lemma 9.2.9: the zag-zag rn'-paths Pan, and P;,o for mm) = 4.

CASE 3. m' = 4. Again, a separate construction is required for the smallest

value of m'. Notice that T' = 3 by (3.8) and hence n' = 11 and c = v. The latter

implies c a 1 (mod 3) so that c 4 {2,3). Let

p0,o = =o u-1 u-5 u2 1111

Creating the paths Po ,j and P;i in the usual way, it is easy to see that the sets

of internal vertices of the Poj , as well as the P<,j, are pairwise disjoint. Hence

Co = $;;: Poj and C,' = P,; are m-cycles. The edge length sets of the

zig-zag Cpaths Po.0 and PG0 are

and

LG = {2,5,8,10).

For i = 1, . . . , - 1 let

and for i = 1,. .., 151 - 1,

Again, the sets of internal vertices of the P,,J ? as well as the P;, , are pairwise disjoint.

Hence C i = Rj and C; = @::: Pi; are m-cycles. The edge length sets of the

zig-zag Cpaths P i , and P:o for i 2 1 are

Li = {1 + l l i , 2 + l l i , 3 + I l ( i - 1),4+ l l i )

and

Lr = ( 5 + l l i , 6 + l l ( i - I), 7 + l l i , 8 + l l i ) .

The sets Li and LT are pairwise disjoint for i 2 0 and we thus have a C,-decomposition

of X(n; L p ) , where L p = (~Li i - l Li ) U (uilfAg1 Ly ). To see that the set LC =

{ I , . . . , "-') 2 - L p satisfies the the conditions of Lemma 3.2.1, we now find the

alternating sums Ai of the leftover edge lengths on intervals (1 + Hi, 11 + Hi], i >_ 0. For c 2 4 we have

A(, = 11,

for i = 1, ..., Lfl -2;

and

if c is odd, and

if c is even.

Again, since k from Lemma 3.2.1 has to be even, several cases arise with respect

to the residue class of c modulo 4.

Subcase e m 0 (mod 4). We have

and

whence

Subcase c = 1 (mod 4), c 2 5. Now

and

so that

Subcase c = 2 (mod 4), c 2 6. We have

and

whence

Subcase c n 3 (mod 41, c 2 7. Now

and

whence

Subcasec= 1. Hereweget Sk = 2 - 3 + 5 - 6 + 8 - 1 0 = - 4 a n d a r + l = 11,sothat

Qk+l - 2sk = 19. Unfortunately, d = 3, whence = 17 and ak+t,l - 2Sk $ so

that we can not use Lemma 3.2.1. However, it is possible to find a central m-cycle

that uses the remaining edge lengths. One such central m-cycle is given by the

equality

2 - 3 + 5 - 6 + 8 - 1 0 + 1 2 - 1 3 + 1 4 - 15+17= 11.

(Observe that this occurrence does not contradict Lemma 3.2.1. Since Co uses an odd number of odd length edges and the total number of odd length edges in K33 is even,

Figure 3.9: Lemma 9.23: the central cycle for m' = 4, c = 1.

there are an odd number of odd edge lengths left over. Hence we need to take 6 = 1

in Lemma 3.2.1 to make EL, (- l)'+'a, - 6 even. Since 4~+1- 2Sk - 26 = 17 5 9, the lemma confirms the existence of a central m-cycle that uses each of the edge

lengths a1 , . . . , a, precisely once.)

We have shown that in all cases X(n; L p ) is C,-decomposable and that the set

of the remaining edge lengths LC = (1,. . . ,?} - L p satisfies the conditions of

Lemma 3.2.1 so that X(n; LC) is C,-decomposable as well.

Hence K, is C,,, -decomposable. O

3.2.4 Conclusion

Lemma 3.2.4 Let n be an odd integer and rn be an even integer such that 6 5 2m 5 n < 3m and n(n - 1) I 0 (mod 2m). Then K,, is C,-decomposable.

PROOF. Let r, d, n', m', r', and c be as defined in Section 3.2.1. If c 5 9, K, is C,-decomposable by Lemma 3.2.2, and if c > 9, K. is C,-decomposable by

Lemma 3.2.3. 0

3.3 Cm-decomposition of K, for n odd, rn even, and m 5 n < 2m

3.3.1 Preliminaries

We have saved the most difficult case for the end. The very basic idea of the

construction is similar to that of the csse m 5 n < 2m for n even, m odd in the

use of central vertices and the rotation p2. However, we now have only one central

vertex and an additional cycle type.

Throughout this section it is assumed that n is an odd integer and rn is an even

integer such that 3 5 m 5 n < 2m and n(n - 1) = 0 (mod 2m).

We shall view the graph K, as the join K,,-l K1, where the vertex set of Knml is {uO,u l , . . , , u , 4 and the vertex of Kl (that is, the central vertex) is w .

We shall now discuss the parameters. As usual, denote the remainder n - m by

r. So r is odd and r 5 m - 1. Since m is even, rn cannot divide v, whence

r > 1, Let d = gcd(m,n - 1). Then d is even and d = gcd(m,r - 1) as well. Write

n - 1 = dn', m = dm', and r - 1 = dr'. Observe that rn' = 1 implies ml(n - 1) and

hence, by the restriction m 5 n < 2m, m = n - 1. But then n(n - 1) $ 0 (mod 2m)

since n is odd. Hence m' > 1.

If K, is C,-decomposable, the number of m-cycles is going to be

t t 1 and this is assumed to be an integer. Since t is odd, = 5 must be an odd

integer. Hence, since gcd(ml, r ') = 1, there exists an odd integer c such that r = cm'. t r 1 Since r is odd, this in turn implies that m' is odd, and since = m' is odd, r'

is odd as well. Therefore n' = rn' + r' is even and n - 1 = dn' r 0 (mod 4). Hence

the diameter length F, denoted by D, is even. Furthermore, since r < m - 1,

c s d - 5 . H e n c e c s d - 1 .

The expression (3.13) for the number of m-cycles now attains the form

Since this expression is of the same form as the expression 2.4 for the number of

m-cycles in the case n even, rn odd, we try to follow the example of the construction

in Section 2.3. So central cycles would be generated by applying the rotation n-l p2i, i = 0 , . . . 7 T - 1, to a central cycle that contains the central vertex - this

would take care of the edges between and Kl. The rest would be peripheral

cycles similar to those of Section 2.3.

This all sounds very well, but how do we take care of the edges of diameter

length y? The problem with these is that there are only of them, rather than

n - 1, the number of edges of each of the other lengths. This problem is solved in

two different ways, depending on whether c 2 f or c < 4 . For c 2 4 a new type of a

cycle, called a diameter cycle, is introduced. The generating diameter cycle contains

two edges of diameter length, one from each of the two orbits of (p2 ) . .4pplying p2i,

n- I 2 = 0,. . .7 -- 4 1, to this cycle we obtain 9 diameter cycles which use up all

the edges of the diameter length. The remaining c$ - "=I - - ( c - 8) $ cycles are

peripheral, similar to those of Section 2.3.

For c < i, however, the number of cycles is not large enough to include 9 diameter cycles on top of the 9 centrd cycles. So edges of diameter length have

to be used up in central cycles. We shall have two generating central cycles which

contain edges of diameter length from distinct orbits of (p2) , one edge for each cycle.

Applying pZi, i = 0,. . . , -- n-I 4 1 to each of them results in the 9 central cycles

we aimed for. Having only one edge of diameter length in a cycle, however, causes

some problems, because it means that the rest of the edge lengths represented in

the cycle cannot be paired up. As we shall see shortly, this can be fixed by "magical

switchesn.

3.3.2 The case c < f 3.3.2.1 Central cycles

In this case, diameters appear in central cycles, which are generated by two slightly

different cycles. Coupled peripheral cycles will be similar to those of Section 2.3.

Solitary peripheral cycles, however, will be the most difficult to construct and will

be quite different from any other peripheral cycles. The construction of solitaty

peripheral cycles splits into two subcases: m' = 3 (mod 4) and m' m 1 (mod 4).

The following two lemmas, however, explain the construction of the two types of

central cycles in both cases. This involves the "magical switches" we have announced

before.

Since c < f , observe that d 2 4. In addition, dr' + 1 = r = cm' < fmt, which m'- 1 implies r' < $ - f . Since rn' is odd, we have rt 5 7.

Figure 3.10: Lemma 3.9.1: the paths RI and R2 that generate the central cycles of the first type.

Lemma 3.8.1 Let LC be a subset of the edge length set (1,. . . , D) with the following

properties:

1. ILcl = 2 - 1,

2. {1,4,D) LC, and

PROOF. Let LC = {1,4,a3,. . .,a?-,, D), where 5 c a:, c . . . < a?-2 < D. Define the zig-zag (7 - 3)-path P by

where ,4 = 4 - a3 + a, - . . . + (-l)y-2aF-2. Furthermore, let

Observe that T contains one edge of diameter length and two diametrically opposed

edges of each of the lengths 4, as,. . . , a?-2. Since D is even, these two diametrically

opposed edges belong to the same orbit of ( p 2 ) .

Now define the paths R1 and Rz by:

Notice that the last three vertices of p(T) are us-.,+D, u i + ~ , and u ~ + 1 Since

a3 1 6, u ~ - . ~ + r , # uo and the vertices of R2 do not overlap.

Observe that IE(RI)I = IE(R2)I = m - 2. Together, R1 and R2 contain exactly

four edges of each of the lengths 1,4, a ~ , . . . , a?-2 (that is, two pairs of diametrically

opposed edges, each pair from a different orbit of ( p 2 ) ) , exactly two edges of diameter

length (from distinct orbits of ( p 2 ) ) , and a pair of diametrically opposed edges of

length 2 from the orbit (p2) (uou2). Therefore

is a decomposition of X(n - 1; LC) $ (p2)(uou2) into (m - 2)-paths.

Define the central m-cycles CF and CF by

C: = W U ~ R ~ U ~ + ~ W and C: = wu&uDw.

Since the endpoints of each of the paths R1 and Rz form a diametrically opposed

pair and the two pairs belong to distinct orbits of ( p 2 ) ,

Figure 3.1 1 : Lemma 3.32: the paths Rl and R2 that genemte the central cycles of the second type.

Lemma 3.3.2 Let LC be a subset of the edge length set (1, . . . , D) with the following

properties:

1. ILcl = t - 2,

2 {1,6,D) C LC, and

3. 2,3,4,5,7 4 LC.

Then ( ~ ( n - 1; LC) $ ($)({ulu3, u1u., u ~ u ~ } ) ) W Kl *S Cm-decomposable.

PROOF. Let LC = {1,6,a3,. . . , a ? - j , D ) , where 7 < a3 < . .. < arp-3 < D. Define

the zig-zag (: - 4)-pat h P by

where A = 6 - a3 + 0 4 - . . . + (-1)9-3a?L-3, and let

As before, we can see that T contains one edge of diameter length and two diarnet-

rically opposed edges of each of the lengths 6, as, . . . , a=-3. 2

Now define the paths Rl and R2 by:

Notice that, since a3 2 8, u-1 and u ~ - l do not belong to T. Hence the vertices of

R1 do not overlap.

Observe that IE(RI) I = IE(R2) I = - 2. In addition, R1 and R2 contain exactly

four edges of each of the lengths 1,6, US, . . . , a? -3 (that is, two pairs of diametrically

opposed edges, each pair from a different orbit of (p2) ), exactly two edges of diameter

length (from distinct orbits of (p2) ), a pair of diametrically opposed edges of length

2 from the orbit (p2 ) ( 1 4 4 , a pair of diametrically opposed edges of length 3 from

the orbit (p2)(u1u4), and a pair of diametrically opposed edges of length 5 from the

orbit (p2) (U I u~). Therefore

is a decomposition of X(n - 1; LC)) $ (p2)({u1u3, ~ 1 ~ 4 , uIu6)) into (m - 2)-paths.

As is the previous lemma, if we now define the rn-cycles Cf and C,C by

C: = W U ~ R ~ U ~ + ~ W and C: = wu&u~w,

we can see that 2i CC D { p ( j): j = l , 2 , 2 = O , ...,F- 1)

is a c,-~iecomposition of ( ~ ( n - 1; L ~ ) $ (p2)({u1u3, Ult(4, u ~ u ~ ) ) ) w K ~ . o

3.3.2.2 Coupled peripheral cycles

In this sect ion we construct 9 families of coupled peripheral cycles that represent

a C,-decomposition of a circulant X(n - 1; LCP).

Lemma 3.3.3 There exists a set L.cp UM the foilolLr'ng properties:

1. L c p {n' + 1,. . . , D - n'},

PROOF. Throughout this proof let a = 9. First we assume that m' 2 5 . Define

the zig-zag (m' - 1)-path by

Notice that this is precisely the same (m' - 1)-path as in case rn' 1 5, r' 5 3 of Lemma 2.3.4. Its edge length set is

For i = 2, . . . , a, obtain the zig-zag (ml- 1)-path PQ from P1,o by adding 2( i - l)nt

to the subscripts of the even vertices. The paths in each of the families Pi = {pJn'(A,o) : j = 0,. . . , d - 1) are thus pairwise vertex-disjoint and the edge length

set of PtVo is Li = L1 + 2(2 - 1)n'.

We shall use the method of Paragraph 1.4.1.1 to connect each family of (m' - 1)-

paths into an m-cycle. Since the terminal vertex of each of the paths PiVo is u,l+l,

an edge of length I in the auxiliary circulant results in connecting edges of length

(I - 1)n' - 1 or ( I + l )nf + 1, depending on the orientation of the cycle.

We first assume that d 2 20. Define a number p as follows. If is odd, let

p = f - 4. If ) is even and is odd, let p = f + 2. If 4 and 3 axe both even, let

p = f+1 . Observethat inallcasesgcd(d,p) = 1 . s incec s $ - I , = I $ - 1 . Hence d 4 + 3 1 ~ < ~ - 3 .

If a P 1 (mod 4), take the auxiliary circulant X = X(d; {1,3 ,4 , . . . , y)) and

decompose it into X(d; {l}), X(d; {3,4}), . . . , X(d; { y, 9)). Use the d-cycle

X(d: (1)) with the orientation resulting in connecting edges of length 2nf + 1. The

set of lengths of all connecting edges is thus

If a 1 2 (mod 4, take X(d; {3,4))$ $X(d; { t , 4 + I))$ X(d; { p } ) and use

the d-cycle X(d; {p) ) with both orientations. Hence

[fa 3 (mod 4), take X(d; {I))$ X ( d ; {3,1))$ $ X(d; { y, +))@ X(d; { p } ) . Use X(d; {p}) with both orientations and X(d; {I)) only with the orientation result-

ing in connecting edges of length 2n' + 1. Hence

If a 0 (mod 4), take X(d; {3,4))$ $X(d; { 4 + 1, : + 2)). Hence

In all cases rnin(Lx) = 2n' - 1 and max(Lx) 5 ( 4 - 2)n' + 1 < D - n'. Now let's look at the small values of d. If d 5 6, a = 0 and the lemma is

vacuous. If a = 1, d 2 8 and we take X(d; (1)) with the orientation resulting in

Lx = (272' + l}.This settles the case d < 10. If d = 12 and a = 2, take X(d; {I))$

X(4 ( 5 ) ) . Use each d-cycle with only one orientation; namely, the one resulting in

Lx = 1272' + 1,4nf - 1). I f d E {14,16), take X(d; {I))$ X(d; {3)), and if d = 18,

take X(d; {I))$ X(d; (5)). In these cases a 5 3. Use X(d; (1)) with the orientation

resulting in connecting edges of length 2n' + 1, and X(d; (3)) or X(d; (5)) with any

one orientation, or with both orientations, depending on whether a = 2 or a = 3. In all cases min(Lx) 2 2n' - 1 and mw(Lx) < (4 - 1)n' = D - n'.

Observe that the zig-zag (m' - 1)-paths defined above use no edges of lengths

f 1 modulo n'. Hence Lx and U:=, Li are disjoint, LCp = (& Li ) U LX has size

am' = y m ' and X(n - 1; LCP) is C,-decomposable. Since max(Lx) < D - n' and

the longest edge in the zig-zag paths has length

the set LCp satisfies Conditions 1 - 4 of the lemma.

Now let m' = 3 so that n' = 4. This construction will be similar to that of

Lemma 2.3.4 for rn' = 3. Let the zig-zag 2-paths and Pi,, be

Their edge length sets are

L1 = (2 + n', - 1 + 2n') and

For i = 2,. . . , obtain Pi,, from by adding 2( i - l )nt to the subscript of the

second vertex, and for i = 2 , . . . , L;J' obtain P:, from Pi,, in the same way. Then

let Pi = (@"'(P;:,~) : j = O ,..., d - 1) and PT = {(p"(co) : j = 0 ,..., d - I}.

The auxiliary circulant to be used for linking the 2-paths into rn-cycles is X = X(d;{1,3,5 ,..., 2LtJ + 1)). Observe that, since 3c = r = d + 1, a = 9. Hence

d n 2 (mod 6) and gcd(d,3) = 1. In addition, if a is even, gcd(d,a + 1) = 1. If a is even, decompose X into X(d; {I)), X(d; {3)), X(d; {5,7)), . . . , X(d ; {a -

1,.+1}), or X(d; (111, X ( 4 {3)), X ( 4 (5,711, * * . , X ( d ; {a-3,a-1}), X(d; (a+l)) , depending on whether is odd or even, respectively. Now use the d-cycles X(d; (1))

and X(d; (3)) with only one orientation; namely, the one resulting in connecting

edges of lengths n' + 1 and 3n' + 1, respectively. Use the d-cycle X(d; {a + 1)) with

both orientations. The set of lengths of the connecting edges is thus

Lx = {n' + 1,3n' + 1) U {f 1 + (22 + l )nf : i = 2,. . . , 4).

If a is odd, take X(d; {I))$ X(d ; {3))$ X(d; ( 5 , ?))$ . . . $X(d ; { a - 2, a } ) , or

X(d; {I))$ X(d; {3,5))$ . . . $ X(d; {a-2, a ) ) , depending on whether a H 3 (mod 4)

or a n 1 (mod 4), respectively. Now use the d-cycle X(d ; (1)) only with the ori-

entation resulting in connecting edges of lenth n' + 1, and X(d; {3)) with both orientations. Hence

1tJ It is easy to see that in both cases Lx and (UiZ1 L i ) U (U+, LT) are disjoint. Let

LCP be their union. The set LCp thus satisfies Conditions 2 - 4 of the lemma.

The minimum element of LCp is clearly n' + 1, while the maximum is at most

since a = 9, n' = 4, and d 2 8. Hence Condition 4 is satisfied as well.

This proves the lemma for all cases. o

In the next two sections we shall construct solitary peripheral cycles and combine

them with the coupled peripheral cycles just described. Before we move on, however,

let us define a path of length 4 that will play an important role in the construction

of solitary peripheral cycles for both m' 3 (mod 4) and rn' m 1 (mod 4).

X - l

Figure 3.12: The crossover &path Q,(z,y).

Definition 3.3.4 A 4-path of the form

denoted by Q, ( x , y ) , is called a crossover 4-path.

The following is an easy observation.

Lemma 3.3.5 if s is odd, Q, (3, y ) contains a pair of edges of each of the two lengths

ly - I + $ 1 and ly - I + 24, and the two edges in the pair belong to distinct orbits

of ( p 2 )

3.3.2.3 Solitary peripheral cycles for rn' 1 3 (mod 4)

Lemma 3.3.6 Let either rn' 3 (mod 8 ) and n' o 2 (mod 4) , or rn' 7 (mod 8)

and n' r 0 (mod 4), or m' = 7 and n' = 10, or rn' = 3. Then t h e n exists a set

L p {I,. . . , D) such that

PROOF. First let us assume that rn' 2 7 and either m' n 3 (mod 8) and n' a 2 (mod 4), or m' 1 7 (mod 8) and n' 0 (mod 4). Let

1 mt-5 = I ( T + n ' + 5 ) and

= -EkE 2 + = f(2nt - m' + 15).

Notice that in both cases 2 is odd and z is even.

r+D

Figure 3.13: Lemma 3.9.6: the mt-path Pogo

mt -5 For j = 1,. . . , d - 1, let Polj = @ " ' ( P ~ ~ ) . Since < n' - - 2 7 the paths

Poi and Po,j+l are in both cases vertex-disjoint except for the endpoints. Since u,

does not belong to PoC, Poj and POj+) are vertex-disjoint, and since I = f(2n' - m'-5 m' + 15) < n' - 7, U+ does not belong to POql SO that Po,j+l and Po,.+

vertex-disjoint. Hence Go = u;G Po,j is an m-cycle.

Observe that Po,o contains exactly two edges of each of the lengths in the set Lo,

where

Lo= {3,5,8,9,12,13 ,..., m ' - 3 , m ' - 2 , D - Z }

if x 5 F. Moreover, the two edges of the same length belong to distinct orbits of

(p2 ) . In addition, contains an edge of length 2 from the orbit (p2 ) (uI 4. Hence

is a C,-decomposition of X(n - 1; Lo) $ (p2)(uIu3).

If c = 1, let L p = Lo. Since (Lol = 2 = 2 7 it is easy to see that conditions

1 - 1 are satisfied. We may thus assume that c 2 3. Now let LCp be a set satisfying the conditions of Lemma 3.3.3 and let L p =

Lo U LCP. Since the maximum element of Lo is

the sets Lo and LCp are disjoint. Now

Since X(n - 1; LCP) is Cm-decomposable by Lemma 3.3.3, it is now easy to see that

the set L p satisfies conditions 1 - 4.

Figure 3.14: Lemma 3.3.6: the mt-path Pop for rn' = 7, n' = 10.

In the case m' = 7, n' = 10, let

With Lo = {3,5,8), the proof can now be completed as before.

Finally, let m' = 3 and hence n' = 4. Let

With Lo = {3). the proof can now be completed as before except that the edge length

5 has to come from the set LCP. Fortunately, a set LCp satisfying the conditions of

Lemma 3.3.3 for m' = 3 does contain 5 whenever c 1 3. Since c = 9 is odd and d 2 4, we indeed have c 2 3. Hence 5 E L p and Conditions 1 - 4 are met for the

case rn' = 3 as well. This proves the lemma for dl cases. 0

Lemma 3.3.7 Let rn' 2 11 and either m' 3 (mod 8) and n' a 0 (mod 4), or

rn' z 7 (mod 8) and n' E 2 (mod 4). Then there ezists a set L p E (1,. . . , D) such

that

3. 1,2,3,5,6,D 4 Lp, and

PROOF. Let

Z = f ( + + n t + 5 ) and mt -1 x = -- 2 + Z = !(2nf - m' + 11).

Notice that in both cases Z is odd and x is even.

Define the m'-path Po,* by

Figure 3.15: Lemma 9.9.7: the m'-path PoVo

i f x s - 2

mt+l < nt - ml-1 For j = 1, .. . , d - 1, let Poj = p ' " ' ( ~ ~ , ~ ) . Since 2 7 the paths Poj and Poj+l are in both cases vertex-disjoint except for the endpoints. Since

u, does not belong to Po,o, POj aad Po,j+t are vertex-disjoint as well, and since

t = L(2nr I - m' + 11) < n' - 9, u, does not belong to Poql so that Po,j+l and

POj+# are vertex-disjoint. Hence Co = (J$i POj is an m-cycle.

Observe that Po* contains exactly two edges of each of the lengths in the set Lo, where

Lo = {4,7,8,12,13,16,17 ,..., m ' - 3 , m ' - 2 , D - 2 )

i f z > * , m d

if x 5 9. The two edges of the same length belong to distinct orbits of (p2 ) . In

addition, Poqo contains one edge of length 2 from the orbit (p2) (uouz), one edge of

length 3 from the orbit (p2 ) (uou3), and one edge of length 5 from the orbit ( p 2 ) (uou5).

Hence n ' {p2'(Co) : i = O? .. .: - 1)

is a C,-decomposition of X(n - 1; Lo) $ (p2)({uouz, uous, uous)).

If e = 1, let L p = Lo. Since ILol = = e, 2 it is easy to see that conditions

1 - 4 are satisfied. We may thus assume that c 2 3.

Now let LCp be a set satisfying the conditions of Lemma 3.3.3 and let L p =

Lo U LCP. Since the maximum element of Lo is

the sets Lo and LCp are disjoint. Now

Since X(n - 1; LCP) is C,-decomposable by Lemma 3.3.3, it is now easy to see that

the set L p satisfies conditions 1 - 4. 0

3.3.2.4 Solitary peripheral cycles for m' = 1 (mod 4)

Lemma 3.3.8 Let either rn' i 1 (mod 8) and n' i 2 (mod 4) and m' 2 17, or

m' E 5 (mod 8) and n' r 0 (mod 4), or m' = 13 and n' = 14, or m' = 5. Then

t h e n ezists a set L p C_ {I,. . . , D ) such that

9. 1,2,4, D 4 L p , and

Figure 3.16: Lemma 3.3.8: the mt-path Po,o.

PROOF. First let us assume that either rn' 1 1 (mod 8) and n' = 2 (mod 1) and

m' 2 17, or rn' = 5 (mod 8) and n' z 0 (mod 4). Notice that, since r' < 9 and r' is odd, the last condition implies rn' 2 13. Let

m'-3 Z = ; ( n t - 5 - T ) , m'-3 x = - + z = L Q n '

2 ( + m f - 13), and

m'- 1 i f y m't3 For j = 1 ,..., d - 1, let Poj PO,^). Since < n' -- 7, m'-3 the paths Poj

and Po,j+l are in both cases vertex-disjoint except for the endpoints. Since u 2

and u-, do not belong to Po,o, Po,j and are vertex-disjoint as well, and since

un,- and u , do not belong to Po$, Po ,j and P,lj+t+l are vertex-disjoint. Hence ;I

Co = u;;: Poj is an m-cycle.

Observe that PoVo contains exactly two edges of each of the lengths in the set Lo, where

Lo= {3,5,8,9,12,13 ,..., m ' - 5 , m ' - 4 , D - Z , D - 1)

if y > v, and

if y 5 e. 2 The two edges of the same length belong to distinct orbits of ( p 2 ) . In addition, Polo contains an edge of length 2 from the orbit ( p 2 ) (u&. Hence

is a C,-decomposit ion of X(n - 1 ; LO) $ ( p 2 ) (uI u3). ml-l r-1 If c = 1, let L p = Lo. Since lLol = 7 = T , it is easy to see that conditions

1 - 4 are satisfied. We may thus assume that c 2 3.

Now let LCp be a set satisfying the conditions of Lemma 3.3.3 and let L p = Lo U LCP. Since the second largest element of Lo is

the sets Lo and LCp are disjoint. Now

Since X(n - 1; LCP) is C,-decomposable by Lemma 3.3.3, it is now easy to see that

the set Lp satisfie conditions 1 - 4.

D+2 D 3

Figure 3.17: Lemma 9.3.8: the m'-path PoBo for m' = 13, n' = 14.

Now let us consider the case m' = 13, n' = 14. W e let

where &(-4,6) is, of course, u - ~ u ~ u - s u ~ u - ~ and Qs(-4, D - 3) is the crossover

Cpath Z I - ~ U ~ + ~ U - ~ U ~ - ~ U - I ~ . With LO = {3,5, 117 12? D - 6, D - 11) the proof can

now be completed as before.

Figure 3.18: Lemma 9.5.8: the m'-path Po,* for m' = 5, n' = 6.

Finally, let m' = 5 and hence n' = 6. Let

so that Lo = {3,5) and complete the proof as before.

This proves the lemma lor all cases.

Lemma 3.3.9 Let either m' s 1 (mod 8) and n' t 0 (mod 4) and m' 2 17, or rn' 1 5 (mod 8) and n' m 2 (mod 4) and m' 2 21, or m' = 13 and n' = 18, or

m' = 9. Then there ezists a set L p C {I , . . . , D) such that

5. 1,2,3,5,6,D Q L p , and

PROOF. First assume that either rn' = 1 (mod 8) and n' = 0 (mod 4) and m' 2 17,

or m' r 5 (mod 8) and n' 1 2 (mod 4) and m' > 21, or m' = 13 and n' = 18. Let

= i(n' - 5 - y), x = v + Z , and

y = n ' - x .

Notice that in all cases Z is odd while z and y an even. Since m' + 2r' 1 23, Z 2 3.

Define the m'-path POvo by

Figure 3.19: Lemma 3.3.9: the m'-path Pope.

if y > 9, and by

mt -5 i f y S T .

For j = 1, . . . , d - 1, let Poj = p'"' (Po*). Since < n' - F, the paths Pigj and Po,j+l are in both cases vertex-disjoint except for the endpoints. Since u ,.,~-l

a and u, do not belong to Po,o, POj and Po,,+) are vertex-disjoint, and since u,, -1-1

1

and u, do not belong to Pop, Poj and Po,j+f+l are vertex-disjoint as well. Hence Co = u$; Poj is an m-cycle.

Observe that Po,o contains exactly two edges of each of the lengths in the set Lo,

where

Lo = {4,7,8,12,13,16,17 ,..., m'-5 ,m'-4 ,D-Z,D-1)

if y 5 9. The two edges of the same length belong to distinct orbits of ( p 2 ) . In addition, contains one edge of length 2 from the orbit (p2)(uouz), one edge of

length 3 from the orbit ( p Z ) (uou3), and one edge of length 5 from the orbit (p2) (uou5).

Hence n ' {p2'(co) : 2 = 0,. . . , q - 1)

is a C,-decomposition of X(n - 1; Lo) $ (p2)({uouz, uou~, uous)). 3 = t - 3 If c = 1, let L p = Lo. Since lLol = "f 2 , it is easy to see that conditions

1 - 4 are satisfied. We may thus assume that c 2 3.

Now let LCp be a set satisfying the conditions of Lemma 3.3.3 and let L p = Lo u LcP Since the second largest element of Lo is

the sets Lo and LCp are disjoint and

Since X(n - 1; LCP) is C,-decomposable by Lemma 3.3.3, it is now easy to see that

the set L p satisfies conditions 1 - 4.

Finally, let rnt = 9 and, since gcd(mt, nt) = 1, n' = 10. Let

With Lo = {4,7,8} the proof is completed as before.

Figure 3.20: Lemma 9.9.9: the m'-path Poto for rn' = 9 and n' = 10.

3.3.3 The case c 2 4 Two slightly different constructions are required depending on whether $ is odd or

even. The following lemma provides the basis for the construction of central cycles

and diameter cycles in both cases.

Lemma 3.3.10 Let L, Lo, and LD be pairwise disjoint subsets of the edge length

set {I, . . . , D - 1 . Let P be a rig-zag path with the edge length set Lo, let Po be

a zig-zag path with the edge length set LD, and let R be a path in h'n-l with the

following properties:

1. the length of every edge of R is in L u Lo U LD:

2. R contains ezactly one edge of each of the lengths in Lo U Lo and this edge

belongs to the same orbit of (p2) as the edge of the same length in P or PD,

9. R contains ezactly two edges of each o j the lengths in L, one from each of the

two orbits of ( p 2 ) ,

5- precisely one of the two endpoints of R has an odd subscript,

6. IE(PD)I = ILD~ = - 1, and

vaaiea with

b odd nrbrnipts

--prr

Figure 3.21: An ezample o/ a diameter cycle.

7. pncisely one of the two endpoints of PD has an odd subscript.

Then ( ~ ( n - 1; L u LD u {D)) $ ( p 2 ) ( ~ ) ) w Kl is Cm-decomposable.

PROOF. First we describe the central cycles. Conditions 1 - 4 imply that

is a partition of the edge set of X(n - 1; L) $ (p2)(P) $ (d)(PD) into (m - 2)-paths.

Let u,, and u, be, respectively, the initial and the terminal vertex of the path R. Define the cycle Cc by

Cc = W U , ~ R U ~ ~ W .

Conditions 4 and 5 now imply that

We now describe the diameter cycles. Let u,, and u, be, respectively, the initid

and the terminal vertex of the path p(PD). Define the cycle CD by

By Condition 6, CD is an rn-cycle and since D is even, Condition 7 implies that the

two edges of diameter length in CD belong to distinct orbits of ( p 2 ) . On the other

hand, the edges in p*+l(%) belong to the same orbit of ( p 2 ) as the corresponding

edges in p(PD) and are diametrically opposed to them. Hence

We thus have a C,-decomposition of ( ~ ( n - 1; L U LD U { D l ) $ ( p 2 ) ( ~ ) ) w Kl.

3.3.3.1 The case $ odd.

First we describe the peripheral cycles. Since 4 is odd, a = f (c - 4) is an integer.

Lemma 3.3.11 There ezists a set Lcp with the j'ooNoun'ng properties:

2. ILcpl = am', and

PROOF. First observe that d = 2 implies c = 1 and hence a = 0. In this case the

lemma is trivially true. Now rn' = 3 implies r' = 1 and 9 = c 2 4, whence d = 2

and c = 1. We may thus assume that d > 6 and mt 2 5. cm'-1 > m' 1 m'-1 Since c 2 f implies rt = - d 2 > 7, we have rt 2 3. For coupled

peripheral cycles we shall use the construction of Lemma 3.3.3 (case m' 2 5) with

a few modifications. We ask the reader to refer to the proof of Lemma 3.3.3 since

only an outline and the differences will be presented here.

First of all, a = 9 of Lemma 3.3.3 is replaced by a = $(c - i). Since c 5 d - 1,

we have a < 9. As in Lemma 3.3.3, the basis for the construction is the zig-zag

(m' - 1)-path

with edge length set

and for i = 2, . . . , a we obtain the zig-zag (m' - 1)-paths Pito with edge length sets

Li = L1 + 2(i - l )nt by adding 2(i - 1)nt to the subscripts of the even vertices in

S . 0

For d 2 14, let p = f - 2. Since 4 is odd, gcd(d,p) = 1. Since a 5 y, = 2 + 3 5 p. We now define the set Lx of connecting edge lengths for the various

values of a modulo 4 as in Lemma 3.3.3. Notice that in all cases min(Lx) = 272' - 1

and max(Lx) 5 ( f - l )nt + 1 < D - 4.

For d 5 10 we have a 5 2. The connecting edge lengyjs are now chosen in the

following way. If a = 1, use the d-cycle X(d; (1)) with the orientation resulting in

Lx = {2nt + 1). If a = 2, we must have d = 10. Now use the d-cycle X(d; (3)) with

both orientations so that Lx = {2nr - 1,472' + 1). In these two cases min(Lx) 2 2nt - 1 and max(Lx) 5 (f - l)n' + 1 < D - 4.

Since Lx and Rt1 Li are disjoint, LCp = (U:=I Li) u LX has size am' and X(n - 1; LCP) is C,-decomposable. Since the longest edge length in the zig-zag paths is

now

the set LCp satisfies Conditions 1 - 3 of the lemma.

Next, the details of the construction of central cycles.

Lemma 5.3.12 Let LC = (1,. . . , D) - LCP, where LCp ia a set satisfying the conditions of Lemma 9.9.1 1. Then X(n - 1; LC) w Kl is C, -decomposable.

PROOF. If m' = 3, we can see that r' = 1, c = 1, d = 2, and a = 0. Since the set

LCP of Lemma 3.3.11 i s now empty, LC = { l , 2,3,4). We now apply Lemma 3.3.10

with Lo = 8, L = {3), LD = {1,2), PD = ~ o u l u - ~ , and R = usuoulu-luz to see

that X(n - 1; LC) Kl is C,-decomposable.

W e may now assume that m' 2 5 and hence r' 2 3. If rn' = 5 , we have r' = 3

and hence d = 1 4. This implies that either m' 2 7 or d 2 6, whence m 2 14.

Condition 1 of Lemma 3.3.11 implies that {D - 2, D - 1, D} C LC. Let LC be the disjoint union of the sets LB, LA, and {D - 1, D) with the property that

max(LB) < min(LA) and lLAl = - 1. That is, LA is the set of the - 1 largest

members of Lc - {D - 1, D) and Ls is the remainder. Since m 2 14,

Hence the sets LB and LA are well defined.

We now introduce the rest of the notation for this proof:

Lg = (1, a* , . . . , a ~ ) , where 2 5 az < . . . < a ~ ,

LA = { a ~ + ~ , . . . , a ~ - l , D - 21, where aN < aN+l < . . . < a ~ - 1 < D - 2,

B = -a2 + as - + ( - l )Nwla,v ,

A2 = B + ( - l ) * e ~ + l + (-l)N+laN+Z + + (-1)M-242M-1, and

A = Az + ( - 1 ) y ~ - 2 ) .

Define the following zig-zag paths:

Since n' is even, each of the a families of coupled peripheral cycles in Lemma 3.3.11

uses an even number of odd edge lengths. Hence LC contains an odd number of odd

edge lengths if and only if f is odd. From the way A is defined it now follows that

A is odd if and only if 9 is odd.

Since [LC - {D - 1, D) I = D - am' - 2 and m' is odd, we have -4 > 0 (evaluated

in the integers) if and only if a is odd. Note that A > 0 means that u~ is in the set

{ul,uz,. . . , u ~ - ~ } while A < 0 means that u~ is in the set { U ~ + ~ , U D + ~ , . . . , We are now ready to define the paths P, PD, and R that satisfy the conditions

of Lemma 3.3.10. The details of the construction depend on whether A - B, a =

i (c - 4 ), and f are odd or even. In all cases, however, the set Lo of Lemma 3.3.10 will be empty so that P is an empty path. Observe that since $ is odd, M - N = " 2 - 1 = tm' - 1 is even. Hence A and B are either both positive or both negative.

Figure 3.22: The path R for Case 1.1.1: A - B odd, a and even.

1. Case A - B odd. Let L = Lg u {D - 1) and let Po = PA SO that Lo = LA. Notice that Conditions 6 and 7 of Lemma 3.3.10 are satisfied for the path PD. We

shall define the path R to meet Conditions 1 - 3. Since 1 LI = 9 by (3.15) and

1 LD 1 = t - 1, Conditions 1 - 3 imply Condition 4. The only thing that requires

verification is the endpoints of the path R and the orbits of the linking edses. which

are the edges of length D - 1, and in some of the cases, edges of length 1.

1.1. Subcase a even. Hence A < 0 and B < 0.

1.1.1. Subcase f even. Thus A is even and B is odd. Let

R = P ~ + ~ ( & ) U D U - ~ P B P A U A U A + D + I

The subscripts of the endpoints of R are B + D + 1 (even) and A + D + 1 (odd).

The orbits of (p2 ) containing the linking edges are:

1.1.2. Subcase f odd. Thus A is odd and B is even. Let

The subscripts of the endpoints of R are B + D + 1 (odd) and A + D - 1 (even).

The orbits of ( p 2 ) containing the linking edges are:

1.2. Subcase a odd. Hence A > 0 and B > 0.

1.2.1. Subcase f even. Thus A is even and B is odd. Let

R = PD-l($)~D-lu-l pB P ~ u ~ z L ~ + ~ - , .

Figure 3.23: The path R for Care 1.1.2: A - B odd, a even, f odd.

Figure 3.24: The path R for Case 1.2.1: A - B and a odd, Q even.

The subscripts of the endpoints of R are B + D - 1 (even) and A + D - 1 (odd). The orbits of (p2) containing the linking edges are:

(P*) ( U D - 2 ~ - I ) = (P~)(P(uOUD-I )) = ( p 2 ) (P(UAUA+D-1 ))=

Figure 3.25: The path R for Case 1.2.2: A - B, a, and f odd.

1.2.2. Subcase f odd. Thus A is odd and B is even. Let

R = P ~ - ' ( & ~ L I U O P B ~ PAUUA+D-~U+DUA+D+I.

The subscripts of the endpoints of R are B + D - 1 (odd) and A + D + 1 (even).

The orbits of ( p 2 ) containing the linking edges are:

Figure 3.26: The path R for Case 2 . I . I : A - B, o and f even.

2. Case A - B even. Since the subscripts of the endpoints of PA are both even

or both odd in this case, we replace the edge of length D - 2 in PA by an edge of

length D - 1. That is, let L = Le U {D - 2) and let PD = P , @ & u ~ ~ + ( - ~ ) M - ~ ( ~ so that LD = (LA -- { D -- 2)) U {D - 1). Conditions 6 and 7 of Lemma 3.3.10 are

thus satisfied. Again, we construct the path R so that Conditions 1 - 3 are met,

and Condition 4 is follows automatically. As before, we'll check the subscripts of

the endpoints of R and the orbits of the linking edges, which are in this case the

edges of length D - 2 and 1, and the orbit of the Umisplaced" edge of PD of length

D - 1.

2.1. Subcase a even. Hence A < 0, B < 0, and A2 = A + (D - 2) > 0.

2.1.1. Subcase f even. Thus A, A2, and B are all even. Let

R = p D + ' ( & ) ~ D + I ~ - I W I p & P A I ~ A ~ - ( D - ~ ) u & + ~ u A ~ + ~

The subscripts of the endpoints of R are B + D + 1 (odd) and A2 + 2 (even). The orbits of (p2) containing the linking edges are:

( p 2 ) ( u ~ 2 + 3 u ~ , + 2 ) = ( P ~ ) ( U O U I ) = ( P ~ ) ( P ( ~ - ~ U * ) )

b 2 ) ( u ~ + l u - 1 ) = ( P ~ ) ( P ( U ~ ~ D - ? ) ) = ( p 2 ) ( P ( u A ~ u & - ( D - ~ ) ) ) , and

( P ~ ) ( u A , - ( D - ~ ) u A , + ~ ) '= ( P ~ ) ( P ( U O U D - ~ 1) = ( P ~ ) ( U A I U A ~ - ( D - ~ ) ) ) ,

which is the same orbit as that of the edge of length D - 1 in PD.

Figure 3.27: The path R for Case 2.1.2: A - B and a even, 4 odd.

2.1.2. Subcase f odd. Thus A, A2, and B are all odd. Let

Observe that since 7 - 1 2 6, we have B + D - 1 > A + D + 1. Hence the vertices

of R do not overlap and the path is well defined.

The subscripts of the endpoints of R are B + D - 1 (even) and A2 + 2 (odd). The orbits of ( p 2 ) containing the linking edges are:

Figure 3.28: The path R for Case 2.2.1: A - B even, a odd, f even.

2.2. Subcase a odd. Thus A > 0, B > 0, and A2 = A - ( D - 2) < 0.

2.2.1. Subcase f even. Thus A, Az, and B are all even. Let

Since - 1 2 6, it is not difficult to see that the vertices of R do not overlap.

The subscripts of the endpoints of R are B + D - 3 (odd) and A2 - 4 (even).

The orbits of ( p 2 ) containing the linking edges are:

Figure 3.29: The path R for Case 2.2.2: A - B even, a and f odd.

2.2.2. Subcase f odd. Hence A, A*, and B axe all odd. Let

R = pD-' ( G ) u D - I u D - Z U O P B ~ P A ~ U A ~ ~ A ~ + ( D - ~ ) U A ~ - ~ U A ~ - ~ -

Since 7 - 1 2 6, it is not difficult to see that the vertices of R do not overlap.

The subscripts of the endpoints of R are B + D - 1 (even) and A2 - 4 (odd).

The orbits of (p2 ) containing the linking edges are:

Since this covers all cases, it now follows from Lemma 3.3.10 that X(n - 1 ; LC) w K1 is C, -decomposable. 0

3.3.5.2 The case f even

d d This case is a natural extension of the case c 2 5, 5 odd, and will be handled in

a very similar way. Since c - f is odd, we will have two types of peripheral cycles:

coupled peripheral cycles, which will be similar to what we had before, and solitary

peripheral cycles.

First we construct the peripheral cycles. Let a = i ( c - f - 1) and observe that

a is ao integer. Since c = 9 2 $ and d 2 4, we have m' 2 5 .

Lemma 3.3.13 Define the rig-zag m'-path PoVo b y

There ezists a set LCp with the following properties:

2. I LcPI = am', and

PROOF. First we construct the solitary m-cycles. The edge length set of PoVo is

Let Polj = pJn'(polo) for j = 1,. . . , d - 1. The longest edge in Poqo, not counting the

edge of length 9 + ($ - l )n t , has length rn' - 1 < n', which means that the paths

Poj are pairwise vertex-disjoint except for the endpoints. Since gcd(d, 4 - 1) = 1, Co = $;:: PoBj is an m-cycle. Hence

{p2'(C0) : 2 = o , . . ., 5 - 1)

is a C,-decomposition of X(n - 1; 0 ) $ ($)(Po$). For coupled peripheral cycles we again use the construction of Lemma 3.3.3 for

m' 2 5 with minor modifications. First of all, we replace a = 9 of Lemma 3.3.3

by a = +(c- - - d : 1). Observe that since c 5 d - 1, a = ) (c - I - 1) 5 4 - 1, which is the same upper bound for a as in Lemma 3.3.3. W e may now closely

follow the construction of Lemma 3.3.3 for m' 2 5, discarding the cases with f odd.

The result is a set LCp satisfying Conditions 1 and 2 of this lemma and with the

property that X(n - 1; Lcp) is C,-decomposable. Since Lo and LCp axe disjoint

and X(n - 1; 0) $ ( p 2 ) (Poqo) is C,,, -decomposable, Condition 3 is satisfied as well.

0

Next, the details of the construction of central cycles.

Lemma 3.3.14 Let the rig-zag m'-path Polo with edge length set Lo be defined as

in Lemma 9.3.13, and let LCp be a set satisfying the conditions of Lemma 9.9.19.

Let LC = (1,. . . , D) - (Lo u LCP) . Then ( ~ ( n - 1; LC) $ (P2)(P-1(~olo)) Kl is C,,, -decomposable.

PROOF. Since the largest member of Lo U LCp is

{D - 2, D - 1, D) LC. Let LC be the disjoint union of the sets LB, LA, and

{D - 1, D ) with the property that mw(LB) < min(LA) and lLAl = 7 - 1. That

is, ILAI is the set of the - 1 largest members of LC - { D - 1, D) and Lg is the

remainder. Now

since d 2 4 and rn' 2 5. Hence the sets Lg and LA are well defined.

The notation we shall use will be similar to that of Lemma 3.3.12:

mf-1 7 + (i - 1)n' ILBIY

ILB( + ILAI = N + - 1,

{a l , a l , . . . , a N } , where rn' = a1 < a2 < . . . < aN,

{ a ~ + l , . . . , a ~ - 1 , D - 21, where a~ < a N + 1 < . . . < O M - 1 < D - 2. -* +a1 - a z + a ~ - --• la^, B + (-l)*aN+l + ( - ~ ) ~ + ' a ~ + ~ + + ( - ~ ) ~ - ~ a ~ - ~ , and

A2 + ( - - I ) ~ - ' ( D - 2).

Define the following zig-zag paths:

We proceed to define the paths P, PD, and R that satisfy the conditions of

Lemma 3.3.10. The details of the construction depend on the residue class of m'

modulo 4 and on whether A - B and a are odd or even. In all cases, however, we

let P = p-I (Po,o).

Observe that since 4 is even, f = $$ is even and M - N = " 2 - 1 = 2 m ' - 1 is

odd. The latter implies that A and B are opposite in sign. Hence A2 and B have

the same sign and

The other inequality that will prove useful in showing that R is indeed a path is

obtained as follows. Since n 5 2m - 1 and n' is even, n' $ 2772' - 2. For d 2 8 we

thus have

d Since 5 is even, D - 2 = $2 - 2 r 2 (mod 4). Hence the number of odd

elements of {I, . . . , D - 2) is odd. Since n' is even, every family of peripheral cycles,

including solitary peripheral cycles, uses an even number of odd edge lengths. Hence

LC - {D - 1, D) contains and odd number of odd edge lengths. This implies that

the alternating sum A is even if m' a 1 (mod 4), and odd if rn' 1 3 (mod 4).

Since lLc - {D - 1, D}I = D - 2 - (a + l)m', A is positive (evaluated in the

integers) if and only if a is even. Note that A > 0 means that u~ is in the set

{ul, u?, . . . , u ~ - ~ ] while A < 0 means that u~ is in the set {uD+1, uo+z,. . . , u ~ ~ - ~ } -

Considering m' to be 1 or 3 modulo 4, and A - B and a to be odd or even

therefore covers all possibilities.

Assume now that d = 4. We have c = 3 and hence a = 0. Since 3m1 = cm' = dr' + 1 = 4r' + 1, we must have rn' n 3 (mod 4). Hence Z is odd.

We now evaluate the alternating sum A for this case:

Hence A is odd and positive, and consequently B is negative.

We have

since m' m 3 (mod 4). Now Ls = {m', 02,. . . , a s ) where either m', az, . . . , a s

are consecutive integers or m', al , . . . ,a,, 2, a;+l , . . . ,a# are consecutive integers for

some i E (2, ..., N - 1). Since m' and Z are both odd and N 1 2 (mod 4), Lg in

either case contains an odd. number of odd edge lengths. From the definition of B we can now see that B is odd. The construction for d = 4 will thus be carried out

in Subcase 2.2.1. For the other seven subcases we may assume that d 2 8.

1. Case A - B odd. Let L = Lg u {D - 1) and PD = PA SO that LD = LA. Notice that Conditions 6 and 7 of Lemma 3.3.10 are satisfied for the path PD . We

define the path R to meet Conditions 1 - 3. Since ILI = f (m - 2m' - 2) by (3.16),

Conditions 1 - 3 imply

E - l = r n - 2 ? IE(R)I = 21LI + lLol+ lLAl = f(m - 2m'- 2) + m'+

so Condition 4 is satisfied as well. For each case we shall verify that the endpoints

of R satisfy Condition 5 and that the vertices of R do not overlap. Note that except

for the alternating sums A, A*, and B, which are evaluated in 2, all calculations

involving subscripts of the vertices are carried out in Z 2 D . We shall also check to

see that the linking edges (that is, the edges of lengths 1. D - 1, and 2) belong to

the appropriate orbits of (p2) .

1.1. Subcase m' r 1 (mod 1). Hence Z = 9 + ( f - l)n' is even.

1.1.1. Subcase a even. Thus A > 0 is even and B < 0 is odd. Notice

A - D - Z + 2 is even. Let

that

Figure 3.30: The path R for Case I . I . I : A - B odd, m' 1 (mod 4, and a even.

Note that the second last vertex in Q1 is and that by (3.17), B - ( A - D + 1) = 3

B - (A2 - 1) > y. Now, since r' > 3,

and by inequality (3.18),

Hence the vertices of R do not overlap and the path is well defined. The subscripts of the endpoints of R are B+ D+1 (even) and - *+A- D- 2+2

(odd). The orbits of ( p 2 ) containing the linking edges are:

1.1.2. Subcase a odd. Thus A < 0 is even and B > 0 is odd. Notice that A + D + Z is even. Let

D-l F j u rnl 1 R = P ( B ; + D - l ~ PBPAUAUA+D-IU+D+Z-I P ~ + ~ + ~ ( Q I ) -

By (3.17), ( A + D - 1)- B = ( A z + l ) - B 2 9. We have

and by (3.18),

'-'- - BLO vcmcer with

Figure 3.31: The path R for Case 1.I.L A - B odd, m' s 1 (mod 4), and a odd.

Hence the vertices of R do not overlap and the path is well defined. The subscripts of the endpoints of R are B + D - 1 (even) and - + A + D + Z

(odd). The orbits of ( P ~ ) containing the linking edges are:

k D + i

A-D

Figute 3.32: The path R for Case 1.2.1: A - B odd, n' 3 (mod 4), a even.

1.2. Subcase m' m 3 (mod 4). Hence Z = 9 + (4 - l)n' is odd.

1.2.1. Subcase a even. Thus A > 0 is odd and B < 0 is even. Notice that A - D- Z is even. Let

R = p D + l ( % ) ~ rn?t +D+lu m F 1 PBPAUAUA-D+IUA-DUA-D-ZP~-~-'(Q~)*

We have

and by (3.18),

Hence the vertices of R do not overlap and the path is well defined.

The subscripts of the endpoints of R are B + D + 1 (odd) and -* + A - D - Z (even). The orbits of (p2 ) containing the linking edges are:

1.2.2. Subcase a odd. Thus A < 0 is odd and B > 0 is even. Notice that A + D + Z is even. Let

We can check that the vertices of R do not overlap exactly as in Subcase 1.1.2. The subscripts of the endpoints of R are B + D - 1 (odd) and - 9 + A + D + Z (even).

The orbits of (p2) containing the linking edges are:

vanica wilb

odd-

* -ruknipo

Figure 3.33: The path R for Case 2.2.2: A - B odd, m' s 3 (mod 4), o odd.

Edges of lhc lm* occurin# in

Figure 3.34: The path R for Case 2.1.1, r' 2 5: A - B even, rn' = 1 (mod 4), a euen.

2. Case A - B even. Since the subscripts of the endpoints of PA are both even

or both odd in this case, we replace the edge of length D - 2 in PA by an edge of

length D - 1. That is, let L = Lg U {D - 2) and let PD = PA2~A1~A2+(-I)~-l(D-I) so that LD = (LA - {D - 2)) u {D - 1). Conditions 6 and 7 of Lemma 3.3.10 are

thus satisfied. Again, we construct the path R so that Conditions 1 - 3 are met,

and Condition 4 is follows automatically. As before, we'll check the subscripts of

the endpoints and the orbits of the linking edges and make sure that the vertices of

R do not overlap.

2.1. Subcase mt G 1 (mod 4). Hence Z = 9 + (d - l )nt is even.

2.1.1. Subcase a even. Thus A > 0, A2 < 0, and B < 0 are all even. First we

assume that r' 2 5. Xotice that Az - Z - 4 is even. Let

We have

since rt 2 5, and by (3.18),

Hence the vertices of R do not overlap and the path is well defined.

The subscripts of the endpoints of R are B + D - 2 (even) and - + A2 - Z - 4

(odd). The orbits of (p2 ) containing the linking edges are:

vatices with

b oddrub#ipa

Figure 3.35: The path R for Case 2.1.1, r' = 3: A - B even, rn' m 1 (mod 4), a even.

NOW let r' = 3. Since dr' + 1 = cm' 2 trn' and d 2 8, we have r' 2 F. This implies m' = 5 and hence n' = 8. We now let

R = pD+'(A)u- m:I +D+l u y 1 P B P A ~

u A ~ uA,+D-~uA~+DuA~+D-~uA~-JuA~ -Z-JpA2-'($3).

Notice that A2 -- Z is even. We have

m'-1 ( A * - 2 - 3 ) - ( A 2 + D ) = - ( D - n ' + + - 3 - D

and by (3.18),

Hence the vertices of R do not overlap and the path is well defined.

The subscripts of the endpoints of R are B + D + 1 (odd) and A2 - Z - 2 (even).

The orbits of ( p 2 ) containing the linking edges have been checked in the case r' 2 5.

2.1.2. Subcase a odd. Thus A < 0, A2 > 0, and B > 0 are dl even. Notice that

A2 + + 2 is even. Let

We have

and by (3.18),

Hence the vertices of R do not overlap and the path is well defined. The subscripts of the endpoints of R are B + D - 3 (odd) and -9 + A2 + Z - 1

(even). The orbits of (p2) containing the linking edges are:

Figure 3.36: The path R for Case 2.1.2: A - B even, m' e 1 (mod 4), a odd.

2.2. Subcase m' n 3 (mod 4). Hence Z = + ( f - 1)n' is odd.

2.2.1. Subcase a even. Thus A > 0, A2 < 0, and B < 0 are all odd.

First assume d > 8 and r' 2 7. Notice that Az - Z - 4 is even. Let

We have

since r' 2 7, and by (3.18),

Hence the vertices of R do not overlap and the path is well defined.

Figure 3.37: The path R for Cuse 2.2.1, d 2 8, r' 2 7: A - B even, m' a 3 (mod 4), a even.

The subscripts of the endpoints of R are B + D - 4 (odd) and - "(itl+ A2 - 2 - 4

(even). The orbits of (p2 ) containing the linking edges and paths are:

NOW let d 2 8 and r' 5 5. Since r' > 9 and rn' r 3 (mod 4), the only

possibility is r' = 5, m' = 7, and hence n' = 12. Let

Notice that A2 -- Z + 2 is even. Now

Hence the vertices of R do not overlap and the path is well defined. The subscripts of the endpoints of R are B + D + 1 (even) and Az - Z - 1 (odd).

The orbits of (p2) containing the linking edges and paths have been verified above.

Figure 3.38: The path R for Case 2.2.1, d 2 8, r' 5 5: A- B even, m' i 3 (mod 4), a even.

Figure 3.39: The path R for Case 2.2.1, d = 4: A - B even, rn' = 3 (mod 4), a even.

Now assume d = 4. Let

Recallthat A = n t - 1 s o t h a t A 2 = A - ( D - 2 ) = n l - 1 - ( 2 n 1 - 2 ) = 3 n t + 1 . Now

Since 2m1 n' + 2,

aN 5 2n' - (n' + 2) - 2 = n' - 4.

On the other hand,

Since also

the vertices of R do not overlap and the path is well defined. The subscripts of the endpoints of R are - + D - B - 2 (odd) and Az - Z - 4

(even). The orbits of (p2 ) containing the linking edges are:

2.2.2. Subcase a odd. Thus A < 0, A2 > 0, and B > 0 are all odd. Let

and

3-10: The path R for Case 2.2.f A - B even, mf s 3 (mod4), odd.

Hence the vertices of R do not overlap and the path is well defined.

The subscripts of the endpoints of R are B+ D - 1 (even) and - +A2 + Z + 1

(odd). The orbits of (p2 ) containing the linking edges are:

We have shown that Conditions 1 - 7 of Lemma 3.3.10 are satisfied for the paths

P, PD, and R in all cases. Therefore ( ~ ( n - 1; LC) $ ( p 2 ) ( p - 1 ( ~ o , o ) ) w Kl is Cm- decomposable. 0

3.3.4 Conclusion

Finally, the pieces of the jigsaw puzzle are put together. For c > $ we show

that the diameter cycles of Lemma 3.3.10, the centrd cycles of Lemma 3.3.12

(Lemma 3.3. PI), and the peripheral cycles of Lemma 3.3.1 1 (Lemma 3.3.13) rep-

resent a C,-decomposition of K,. For c < : we show that the central cycles of

Lemma3.3.1 (Lemma3.3.2) and the peripheral cycles of Lemma3.3.6 or Lemma3.3.8

(Lemma 3.3.7 or Lemma 3.3.9) represent a C,-decomposition of K,, .

Lemma 3.3.15 Let n be an odd integer and m be an even integer such that 3 < m 5 n < 2m and n(n - 1) t 0 (mod Zm). Then K. is C,-decornposabk.

PROOF. Define the parameters t , d, m', n', r', c, and D as in Section 3.3.1. Observe

that K, is isomorphic to X(n - 1; L) w Kl , where L = (1, . . . , D ) .

First assume that c 2 $. If 4 is odd, let LCP be a set satisfying the conditions o

151

f Lemma 3.3.11 and let

LC = L - LCp. Since X(n - 1; LCP) is C,-decomposable by Lemma 3.3.11 and

X(n - 1; LC) K1 is Cm-decomposable by Lemma 3.3.12, X(n - 1; L) w Kl is

C,-decomposable.

For $ even, let the zig-zag m'-path Poto with edge length set Lo be defined

as in Lemma 3.3.13, let LCp be a set satisfying the conditions of Lemma 3.3.13

and let LC = L - (Lo U LCP). Then X(n - 1; L) w Kl can be partitioned into

X(n - 1; LCP) $ ( P ~ ) ( P ~ , ~ ) , which is Cm-decomposable by Lemma 3.3.13, and ( ~ ( n - 1 ; LC) $ (p2) (P- l ( ~ 0 , ~ ) ) ) w K*, which is C,-decomposable by Lemma 3.3.14.

Now let us look at the case c < 6. If (m' n 3 (mod 8) and n' 2 (mod 4)) or (m' n 7 (mod 8) and n' 0 (mod 4))

or (m' = 7 and n' = 10) or m' = 3, let L p be the edge length set from Lemma 3.3.6.

If (m' E 1 (mod 8) and n' 1 2 (mod 4) and rn' 2 17) or (m' E 5 (mod 8) and

n' z 0 (mod 4)) or (m' = 13 and n' = 14) or rn' = 5, let L p be the edge length set

from Lemma 3.3.8. In both cases, X(n - 1; Lp) $ ( p 2 ) ( ~ 1 ~ 3 ) is C,-decomposable.

Furthermore, LC = L - ( L p U {2)) satisfies the conditions of Lemma 3.3.1 so that

(X(n - 1; LC) $ (p2)(uouz)) W Kl is C,-decomposable. Hence X(n - 1; L) w Kl is

Cm-decomposable.

If m' 1 11 and either (m' i 3 (mod 8) and n' n 0 (mod 4)) or (n' = 7 (mod 8)

and n' = 2 (mod 4)), let Lp be the edge length set from Lemma 3.3.7. If (m' s 1

(mod 8) and n' n 0 (mod 4 ) and m' 2 17) or (m' 5 (mod 8) and n' e 2 (mod 4)

and rn' 2 21) or (m' = 13 and n' = 18) or m' = 9, let Lp be the edge length set

from Lemma 3.3.9. In both cases, X(n - 1; L p ) $ (p2)({uou2, uou~, uous)) is C,- decomposable and LC = L - ( L p u {2,3,5)) satisfies the conditions of Lemma 3.3.2 so that (X(n - 1; LC) $ ( P ~ ) ( { u ~ u ~ , ulu4, ulug})) W Kl is C,-decomposable. Hence

X(n - 1; L) w Kt is C,-decomposable.

This covers all cases. Therefore K, is C,-decomposable. 0

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