Discrete Mathematics 318 (2014) 71–77

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Discrete Mathematics

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On the metamorphosis of a G-design into a (G − e)-designMatthewWilliam SuttonSchool of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

a r t i c l e i n f o

Article history:Received 17 December 2012Received in revised form 12 November2013Accepted 18 November 2013Available online 5 December 2013

Keywords:Cyclic graph decompositionLabellings of graphsMetamorphosis

a b s t r a c t

A G-design of order v is an edge disjoint decomposition of Kv into copies of the graph G.A metamorphosis of a G-design of order v into a (G − e)-design of order v is obtained byretaining the graph G−e from each block of G in the design, and rearranging the remainingedges to form further copies of G− e. Here, we prove that if a graph Gwith n edges admitsan α-labelling and the graph G− e admits a ρ+-labelling, then there is a metamorphosis ofa G-design of order 2n(n−1)x+1 into a (G− e)-design of the same order for all integers x.

Crown Copyright© 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction

The concept of a metamorphosis was first introduced by Lindner in 2000 [10]. Since then, there have been several papersdealing with the metamorphosis of G-designs into H-designs. In comparison the study of graph labellings is an older andmore well-known area of design theory; there are over 200 papers published on graceful labellings alone. In this paper, wewill provide a link between these two areas of design theory.

Let K be a simple graph on v vertices. As usual, we denote the vertices of K by V (K) and the edges by E(K). Let Zn denotethe group of integers modulo n. Let V (K) = Zv and let G be a subgraph of K . A G-decomposition of K is an ordered pair (V , B)where V is the vertex set of K and B = {B1, B2, . . . , Bb} is a collection of isomorphic copies of Gwhich partitions the edges ofK . In this case we say that G divides K andwe refer to the copies of G in B as the blocks of the decomposition. A decompositionis said to be cyclic if, for any block Bi in B, the graph obtained by applying the isomorphism i → i + 1 to V (Bi) is also in B.This isomorphism is known as clicking the graph. If K is the complete graph of order v, Kv , then the decomposition, (V , B),is known as a G-design of order v. We also define a (G,H | a, b)-decomposition of a graph K to be the ordered pair (V , B)where V is the vertex set of K and B is a collection of a isomorphic copies of G and b copies of H which partitions the edgesof K . Rosa [11], and El-Zanati, Vanden Eynden and Narong Punnim [5,4], introduced several labelling techniques as tools toconstruct G-designs.

Suppose that G is a simple graph with n edges and no isolated vertices. An injective function h : V (G) → Z+∪ {0} is

called a labelling of G. Let h be a labelling of G and let h(V (G)) = {h(u) : u ∈ V (G)}. Define a function h : E(G) → Z+

by h(e) = |h(u) − h(v)|, where e = {u, v} ∈ E(G). Let h(E(G)) = {h(e) : e ∈ E(G)}. We denote the set of integers{n, n + 1, . . . ,m − 1,m} by [n,m], when n < m; we also define i + [n,m] to be [n + i,m + i].

Consider the following list of conditions:

(1) h(V (G)) ⊆ [0, 2n];(2) h(V (G)) ⊆ [0, n];(3) h(E(G)) = {x1, x2, . . . , xn}, where for each i ∈ [1, n] either xi = i or xi = 2n + 1 − i;(4) h(E(G)) = [1, n].

E-mail address:matthew.sutton1@uqconnect.edu.au.

0012-365X/$ – see front matter Crown Copyright© 2013 Published by Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.disc.2013.11.014

72 M.W. Sutton / Discrete Mathematics 318 (2014) 71–77

If, in addition, G is bipartite with bipartition {A, B} of V (G) (so with every edge in G having one vertex in A and the otherin B), consider the conditions:(5) for each {a, b} ∈ E(G) with a ∈ A and b ∈ B, h(a) < h(b);(6) there is some positive integer λ such that for each a ∈ A and b ∈ B, h(a) ≤ λ and h(b) > λ.

(We remark that here (6) implies (5), but we include both for easy statements of the definitions below.)If a labelling for G satisfies (1) and (3) it is called a ρ-labelling. If G is bipartite and its labelling satisfies (1), (3) and (5)

then the labelling is said to be ordered (a ρ+-labelling). An α-labelling is a labelling for a bipartite graph that satisfies (2), (4)and (6). For a dynamic survey of general graph labellings we refer to [7]. The following theorems are well-known and willbe used in the main results section.

Theorem 1.1 (El-Zanati et al. [5]). Let G be a bipartite graph with n edges. If G has a ρ+-labelling, then there exists a cyclicG-decomposition of the complete graph K2nx+1 for all positive integers x.

Theorem 1.2 (Rosa [11]). Let G be a bipartite graph with n edges. If G has an α-labelling, then there exists a cyclicG-decomposition of the complete graph K2nx+1 for all positive integers x.

Theorem 1.3 (Rosa [11]). Let G be a graph with n edges. There exists a cyclic G-decomposition of the complete graph K2n+1 if andonly if G has a ρ-labelling.

Wenowdescribe ametamorphosisof aG-decomposition ofKv (aG-design of orderv) into a (G−e,H | a, b)-decompositionof Kv . Let G be a simple graphwith n edges and letH be a simple graphwithm edges.We denote by G−e the graph producedby deleting edge e from G. Let (V , B) be a G-design of order v. For each of the a copies of G in B, retain a subgraph G − e,placing each subgraph in the set P(B) to form a partial (G − e)-decomposition (V , P(B)), and place the removed edges inF(B). If the collection of removed edges, F(B), can be rearranged to form b copies of H where each copy is placed in F∗(B),then (V , P(B) ∪ F∗(B)) is a metamorphosis of the G-design of order v into a (G − e,H | a, b)-decomposition of Kv . In thecase that H is isomorphic to G − e, we have a metamorphosis of a G-design of order v into a (G − e)-design of order v.

Several papers have dealt with themetamorphosis of a G-design into anH-designwhere the graphs G andH are specifiedand H is any particular subgraph of G. When the graph G is K4 and the graph H is some subgraph of K4 such as C4, C3 orΘ(1, 2, 2) = K4 − e, the metamorphosis of a G-design into an H-design has been shown to exist for the appropriate orders(see [9,10,8]). We will consider here the existence conditions for the metamorphosis of a G-design into a (G − e)-designwhere G is not explicitly given but belongs to a family of graphs.

In Section 2, we examine the necessary conditions for the existence of a G-design with a metamorphosis into a (G − e)-design and determine a set of admissible orders. Section 3 contains an important theorem, fromwhich themain result followsas a corollary. In Section 4 we present some examples and corollaries of the main result. We conclude with some finalremarks in Section 5.

2. Necessary conditions

Suppose G is a simple graph with n edges, where n ≥ 1, and let d be the greatest common divisor of the degrees of thevertices in G. There are three well-known necessary conditions for the existence of a G-design of order v:(i) |V (G)| ≤ v;(ii) v(v − 1) ≡ 0 (mod 2n);(iii) (v − 1) ≡ 0 (mod d).We will consider the necessary conditions (i) and (ii) above for a G-design of order v (assuming that (iii) holds). Thus theadmissible set of orders contains all v such that v ≥ |V (G)| and

v(v − 1) ≡ 0 (mod 2n).It is simple to check (and well known) that this equation has solutions v ≡ 0, 1 (mod 2n) if n is even and v ≡ 0, 1 (mod n)if n is odd, although there may be other solutions. Now consider the necessary conditions (i) and (ii) for the graph G − e.We find that the set of admissible orders for a (G − e)-design of order v include those v where v ≡ 0, 1 (mod (n − 1))if n is even and v ≡ 0, 1 (mod 2(n − 1)) if n is odd. For a metamorphosis of a G-design of order v into a (G − e)-designwe require the intersection of the set of admissible orders for a G-design and the set of admissible orders for a (G − e)-design. Noting that n2

= (n − 1)(n + 1) + 1 and (n − 1)2 = n(n − 2) + 1 we see that this leads to the set of integersv ≡ 0, 1, n2, (n − 1)2 (mod 2n(n − 1)) which satisfy the necessary conditions (i) and (ii) for a metamorphosis (again theremay be other solutions). In what follows we will consider the case where v ≡ 1 (mod 2n(n − 1)) and describe a methodfor the construction of a metamorphosis of a G-design of order v into a (G − e)-design of the same order v, when G is anybipartite graph with an α-labelling and G − e has a ρ+-labelling.

3. Main result

Theorem 3.1. Let G be a bipartite graph with n edges, and let H be a bipartite graph with m edges. Suppose G has an α-labellingandH has aρ+-labelling. Then there is ametamorphosis of a G-design of order 2nm+1 into a (G−e,H | m(2nm+1), (2nm+1))-decomposition of Kv .

M.W. Sutton / Discrete Mathematics 318 (2014) 71–77 73

Proof. Let h be a ρ+-labelling of H and g be an α-labelling of G. The proof will consist of two parts: the first is a constructionof a G-design of K2nm+1 and the second is a construction of the metamorphosis. The proof is similar to the methods appliedin [5,3].Part 1:Wewill start by constructing a graph G∗ with nm edges such that G divides G∗ and G∗ has a ρ-labelling. Let the vertexset of G have bipartition {A, B} and consider m disjoint copies of B called B1, B2, . . . , Bm. If b ∈ B let bi be the correspondingelement of Bi. Let G∗ be the graph with vertex set A∪B1 ∪ · · ·∪Bm and edge set {abi | 1 ≤ i ≤ m, a ∈ A, b ∈ B, ab ∈ E(G)}.Clearly, G∗ has nm edges and G∗ decomposes into edge-disjoint copies of G.

Recall that h is the ρ+ labelling of H with h(e) = |h(u) − h(v)| where e = {u, v} ∈ E(H) and satisfying conditions 1, 3and 5. By condition 3, h(E(H∗)) = {d1, d2, . . . , dm}, where for each i ∈ [1,m] either di = i or di = 2m + 1 − i.

Recall that g is an α-labelling of G (satisfying conditions 2, 4 and 6). Define a new function, g∗: V (G∗) → [0, 2nm], by

g∗(a) = g(a) if a ∈ A and g∗(bi) = g(b) + (di − 1)n for every bi ∈ Bi where di ∈ h(E(H)).We now show that g∗ is injective. First note that for any vertex v ∈ G∗ we have 0 ≤ g∗(v) ≤ n+ (2m− 1)n = 2nm, and

g∗(A) = g(A) ⊆ [0, n],g∗(Bi) = g(B) + (di − 1)n;

⊆ [1, n] + (di − 1)n= [1 + (di − 1)n, din].

The sets g∗(A) and g∗(Bi) could only intersect if di = 1; however this would contradict the fact that g is an α-labelling.Additionally, the sets g∗(Bi) and g∗(Bj) only intersect if di = dj and this only occurs when i = j. Thus g∗ is injective andg∗(V (G)) ⊆ [0, 2nm] so g∗ satisfies labelling condition 1. It remains to show that the integers |g∗(u) − g∗(v)| are distinctmodulo (2nm + 1) over all ordered pairs (u, v) where {u, v} ∈ E(G∗).

Define g∗(E(A ∪ Bi)) = {|g∗(bi) − g∗(a)| | {a, bi} ∈ E(G∗) and a ∈ A, bi ∈ Bi}. Using the definition of g∗,

g∗(E(A ∪ Bi)) = {|g(b) + (di − 1)n − g(a)| | {a, b} ∈ E(G)}

= {|g(b) − g(a)| | {a, b} ∈ E(G)} + (di − 1)n.

Since g is an α-labelling,

g∗(E(A ∪ Bi)) = [1, n] + (di − 1)n = [(di − 1)n + 1, din].

Now, either di = i or di = 2m+1−i, so either g∗(E(A∪Bi)) = [(i−1)n+1, in] or g∗(E(A∪Bi)) = [2nm−in+1, 2nm−in+n].Notice that

2nm − in + n ≡ −((i − 1)n + 1) (mod 2nm + 1),2nm − in + n − 1 ≡ −((i − 1)n + 2) (mod 2nm + 1),...

2nm − in + 1 ≡ −in (mod 2nm + 1).

It follows that [2nm − in + 1, 2nm − in + n] ≡ −[(i − 1)n + 1, in] (mod 2nm + 1).So regardless of whether di = i or di = 2m + 1 − i, we can consider the set g∗(E(A ∪ Bi)) as covering the integers

[(i − 1)n + 1, in]. Varying i throughout [1,m] forces g∗(E(A ∪ Bi)) = [(i − 1)n + 1, in] to vary throughout [1, nm]. Sothe integers |g∗(u) − g∗(v)| are distinct modulo (2nm + 1) for every edge {u, v} ∈ E(G∗) so g satisfies condition 3 for aρ-labelling. Since g∗ satisfies conditions 1 and 3, g∗ is a ρ-labelling of G∗ and consequently, by Theorem 1.3, there is a cyclicG-design of order 2nm + 1.Part 2: Let e = {a, b} ∈ E(G), where a ∈ A, b ∈ B, be the edge of G that will be removed to form the metamorphosis. Foreach set of vertices, A ∪ Bi ∈ V (G∗), remove the edge ei where ei = {a, b + (di − 1)n}. The remaining graphs form a partialG − e decomposition of K2nm+1. Notice also that the edges removed from G∗ have labels g∗(ei) = |g(b) + (di − 1)n − g(a)|where g(b) > g(a) and di > 0, so g∗(ei) = (g(b) − g(a)) + (di − 1)n.

Let the vertex set of H have bipartition {Y , Z}, where h(z) > h(y) for z ∈ Z, y ∈ Y . We now define a labelling h#

of H such that the edge labels h#(E(H)) induced by h# exactly cover the labels of the removed edges from G∗. Defineh#

: V (H) → [0, 2nm] by h#(y) = h(y)n if y ∈ Y and h#(z) = (h(z) − 1)n + (g(b) − g(a)) if z ∈ Z . We now showthat the edge labels h#(E(H)) match the labels of the edges removed from G∗:

h#(E(H)) = {|h#(z) − h#(y)| | {z, y} ∈ E(H) and z ∈ Z, y ∈ Y }

= {|(h(z) − 1)n + (g(b) − g(a)) − h(y)n| | {z, y} ∈ E(H)}

= {(h(z) − h(y) − 1)n | {z, y} ∈ E(H)} + (g(b) − g(a))= {(di − 1)n | 1 ≤ i ≤ m} + (g(b) − g(a)).

Hence the edges removed from G∗ form copies of H , giving our metamorphosis of the G-design of order 2nm + 1 into a(G − e,H | m(2nm + 1), (2nm + 1))-decomposition. �

74 M.W. Sutton / Discrete Mathematics 318 (2014) 71–77

Fig. 1. Notation for C6 and Θ(1, 3, 3).

Fig. 2. α-labelling of Θ(1, 3, 3) and ρ-labelling of C6 .

The following is a direct result of the theorem proved above.

Corollary 3.2. Let G be a bipartite graph with n edges. Suppose G has an α-labelling and G − e has a ρ+-labelling. Then there isa metamorphosis of a G-design of order 2n(n − 1)x + 1 into a (G − e)-design of the same order for all positive integers x.

Proof. Given that G− e has a ρ-labelling, it follows from Theorem 1.1 that there is a graph (G− e)∗ withmx edges (where xis any positive integer) such that G − e divides (G − e)∗ and (G − e)∗ has a ρ+-labelling. Let the graph H in Theorem 3.1 bethe graph (G − e)∗. It follows that there is a metamorphosis of the G-design of order 2n(n − 1)x + 1 into a (G − e)-designof the same order for all positive integers x. �

4. Examples and corollaries

A Θ(a, b, c) is a cycle of length b+ c with a path of length a connecting two vertices of distance b (or c) apart around thecycle [1]. Here we will consider a metamorphosis of a G-design into a (G − e)-design, where G is the graph Θ(1, k, k) andG − e is the cycle of length 2k (denoted by C2k).

In the following examples the cycle of length 2k on vertices x1, x2, . . . , x2k with edge set {{xi, xi+1} | 1 ≤ i ≤ 2k − 1} ∪

{x1, x2k} will be denoted by (x1, x2, . . . , x2k) or any cyclic shift thereof. A Θ(1, k, k) graph with edge set {{xi, xi+1}|1 ≤

i ≤ 2k − 1} ∪ {x1, x2k} ∪ {x1, xk+1} will be denoted by [x1, x2, . . . , xk−1; xk, . . . , x2k−1, x2k] (or by any of [x1, x2k, . . . , xk+1;

xk, . . . , x2] or [xk, xk+1, . . . , x2k; x1, x2, . . . , xk−1] or [xk, xk−1, . . . , x2; x1, x2k, . . . , xk+1]). The notation is illustrated in Fig. 1.

Example 4.1. Let G = Θ(1, 3, 3) and H = C6. We give a metamorphosis of a Θ(1, 3, 3) design (taking n = 7 and x = 1) oforder (2 × 7 × 6) + 1 = 85 into a 6-cycle system of the same order.

We have an α-labelling [1, 6, 2; 3, 0, 7] of the graph Θ(1, 3, 3) and a ρ+-labelling (1, 3, 2, 7, 0, 4) of the graph C6. (SeeFig. 2.)

Using Theorem 1.1 we can generate six Θ(1, 3, 3) starters to give a Θ(1, 3, 3) decomposition of order 85 as shown inFig. 3.

Remove the dark edge from each of the Θ(1, 3, 3) starters and rearrange them to give the starter shown in Fig. 4 for themetamorphosis.

Corollary 4.2. If k is odd, there is a metamorphosis of a Θ(1, k, k) design of order 2(2k + 1)(2k)x + 1 into a C2k-cycle systemfor all positive integers x.

Proof. El-Zanati et al. [5] showed that there is an ordered ρ+-labelling of C2k for all k where k is odd. In [2] Blinco gave anα-labelling of Θ(1, n,m) where n andm are both positive odd integers withm+ n ≡ 2 (mod 4). If n = m = k, this reducesto an α-labelling of Θ(1, k, k) for all odd k. The result follows from Corollary 3.2. Given below are the required labellings.

M.W. Sutton / Discrete Mathematics 318 (2014) 71–77 75

Fig. 3. Six starters for a Θ(1, 3, 3) design of order 85.

Fig. 4. Single starter for the metamorphosis into a 6-cycle system.

Let the vertices of C2k be 1, 2, . . . , 2k, where i and j are adjacent whenever j ≡ i ± 1 (mod 2k). Define h : V (C2k) →

Z+∪ {0} by,

h(i) =

(i − 1)/2 i odd,2k − i/2 i even, 0 < i < k − 1,2k − 1 − i/2 i even, k − 1 ≤ i < 2k,2k + 1 i = 2k.

Then h is a ρ-labelling of C2k.Let the vertices of Θ(1, k, k) be 1, 2, . . . , 2k, where vertices i and j are adjacent if j ≡ i± 1 (mod 2k) and when: i =

k+72

and j =3k+72 , for k ≡ 1 (mod 4); and i =

k+32 and j =

3k+32 , for k ≡ 3 (mod 4). We define g : V (Θ(1, k, k)) → Z+

∪ {0} asfollows,

g(i) =

(i − 1)/2 i odd,2k + 2 − i/2 i even, 2 ≤ i ≤ k + 1,2k − i/2 i even, k + 3 ≤ i ≤ 2k.

Then g is an α-labelling of Θ(1, k, k).We remark that the above labelling holds for all k ≥ 7. For the small cases k = 3 or 5, an appropriate labelling may be

found in [2].The metamorphosis consists of the cycles of Θ(1, k, k) without the diagonal edge {i, j} where i =

k+72 and j =

3k+72 , for

k ≡ 1 (mod 4); and i =k+32 and j =

3k+32 , for k ≡ 3 (mod 4). These edges are then used completely in starters formed from

the following labelling,

h#(i) =

((i − 1)/2)(2k + 1) i odd,(2k − i/2 − 1)(2k + 1) + d i even, 0 < i < k − 1,(2k − 2 − i/2)(2k + 1) + d i even, k − 1 ≤ i < 2k,2k(2k + 1) + d i = 2k.

Here d =3k+72 −

k+72 for k ≡ 1 (mod 4) and d =

3k+32 −

k+32 for k ≡ 3 (mod 4). �

Example 4.3. An example of Corollary 4.2 (when k = 7, x = 1): We present a metamorphosis of a Θ(1, 7, 7) design oforder 2(15)(14) + 1 = 421 into a 14-cycle system of the same order.

Recall thenotation fromFig. 2. From the labellings of [5,2]wehave anα-labelling [2, 13, 3, 12, 4, 9, 5; 8, 6, 7, 0, 15, 1, 14]ofΘ(1, 7, 7) and a ρ+-labelling, (2, 10, 3, 9, 4, 8, 5, 7, 6, 15, 0, 13, 1, 12), of C14. Here the edge labels of C14 are h(E(C14)) =

76 M.W. Sutton / Discrete Mathematics 318 (2014) 71–77

{1, 2, 3, . . . , 13, 15} so the Θ(1, 7, 7) design of order 421 is given by (V , B) where V = Z421 and B contains the copies ofΘ(1, 7, 7) arising from the following blocks cycled modulo 421:

{[2, 13, 3, 12, 4, 9, 5; 8, 6, 7, 0, 15, 1, 14], [2, 28, 3, 27, 4, 24, 5; 23, 6, 22, 0, 30, 1, 29],[2, 43, 3, 42, 4, 39, 5; 38, 6, 37, 0, 45, 1, 44], [2, 58, 3, 57, 4, 54, 5; 53, 6, 52, 0, 60, 1, 59],[2, 73, 3, 72, 4, 69, 5; 68, 6, 67, 0, 75, 1, 74], [2, 88, 3, 87, 4, 84, 5; 83, 6, 82, 0, 90, 1, 89],[2, 103, 3, 102, 4, 99, 5; 98, 6, 97, 0, 105, 1, 104], [2, 118, 3, 117, 4, 114, 5; 113, 6, 112, 0, 120, 1, 119],[2, 133, 3, 132, 4, 129, 5; 128, 6, 127, 0, 135, 1, 134], [2, 148, 3, 147, 4, 144, 5; 143, 6, 142, 0, 150, 1, 149],[2, 163, 3, 162, 4, 159, 5; 158, 6, 157, 0, 165, 1, 164], [2, 178, 3, 177, 4, 174, 5; 173, 6, 172, 0, 180, 1, 179],[2, 193, 3, 192, 4, 189, 5; 188, 6, 187, 0, 195, 1, 194], [2, 223, 3, 222, 4, 219, 5; 218, 6, 217, 0, 225, 1, 224]}.

To form the metamorphosis, retain a 14-cycle subgraph of each Θ(1, 7, 7) in B, placing each subgraph in the set P(B) toform a partial 14-cycle system (V , P(B)), and place the removed diagonal edges in F(B).

Now rearrange the removed edges in F(B) to form the following 14-cycle:

(30, 141, 45, 126, 60, 111, 75, 96, 90, 216, 0, 186, 15, 171),

and place in F∗(B) the graphs generated by cycling this 14-cycle modulo 421. It follows that (V , P(B) ∪ F∗(B)) is ametamorphosis of this Θ(1, 7, 7) design of order 421 into a 14-cycle system of order 421.

In addition to the corollary above, we present the following corollary to further highlight ourmain result. A kayak paddle,KP(k,m, l), is the graph consisting of two cycles Ck and Cm connected by a path of length l (see [6]).

Corollary 4.4. There is a metamorphosis of a KP(4k, 4k, 1) design of order 2(4k)(8k + 2)x + 1 into a C4k-cycle system for allpositive integers x.

Herewewill apply Theorem3.1withG = KP(4k, 4k, 1),G−e = C4k∪C4k andH = C4k. To apply Theorem3.1wewill requirean α-labelling of the kayak paddle KP(4k, 4k, 1) and a ρ+ labelling of C4k. In [11], Rosa gave the following α-labelling of C4k.Let the vertices of C4k be 1, 2, . . . , 4k, where i and j are adjacent whenever j ≡ i±1 (mod 4k). Define h : V (C2k) → Z+

∪{0}by

h(i) =

(i − 1)/2 i odd, 1 ≤ i ≤ 4k − 14k + 1 − i/2 i even, 2 ≤ i ≤ 2k,4k − i/2 i even, 2k + 2 ≤ i ≤ 4k.

Then h is an α-labelling of C4k. Since h satisfies the α-labelling conditions it will also satisfy the ρ+-labelling conditions. Thuswe can take h as the ρ+ labelling of C4k.

We now construct an α-labelling of KP(4k, 4k, 1) where KP(4k, 4k, 1) consists of the two 4k-cycles, C and C ′, connectedby a path of length 1. Let G be a graph with vertex set {x1, x2, . . . , x4k, y1, y2, . . . , y4k} and edge set {{xi, xi+1}, {yi, yi+1} |

1 ≤ i ≤ 4k−1}∪ {x1, x4k}∪ {y1, y4k}∪ {x4k−1, y4k}. Clearly G has 8k+1 edges and is the union of two cycles one on verticesx1, x2, . . . , x4k and the other on vertices y1, y2, . . . , y4k connected by the edge {x4k−1, y4k}. We denote the cycle on verticesx1, . . . , x4k by C and the cycle on vertices y1, . . . , y4k by C ′. It follows that G is a KP(4k, 4k, 1). Let g : V (G) → N be thelabelling defined below. We claim that g is an α-labelling.Labelling C: First label the vertices x1, x2, . . . , x4k using a variation on Rosa’s labelling,

g(xi) =

(i − 1)/2 i odd, 1 ≤ i ≤ 4k − 18k + 2 − i/2 i even, 2 ≤ i ≤ 2k,8k + 1 − i/2 i even, 2k + 2 ≤ i ≤ 4k.

The labelling gives vertex labels g(xi) = [0, 2k − 1] if i is odd and g(xi) ⊆ [6k + 1, 8k + 1] if i is even and edge labelsg(V (C)) = [4k + 2, 8k + 1].Labelling C ′: This time the labelling is just a cyclic shift of Rosa’s labelling. Label the vertices y1, y2, . . . , y4k by

g(yi) =

2k + (i − 1)/2 i odd, 1 ≤ i ≤ 4k − 16k + 1 − i/2 i even, 2 ≤ i ≤ 2k,6k − i/2 i even, 2k + 2 ≤ i ≤ 4k.

This gives vertex labels g(yi) = [2k, 4k − 1] if i is even and g(yi) ⊆ [4k, 6k] if i is odd and edge labels g(V (C ′)) = [1, 4k].Consider the edge joining C and C ′, e = {x4k−1, y4k} ∈ E(G), we have, g(e) = |6k − 2k + 1| = 4k + 1. Clearly

g(V (G)) ⊆ [0, 8k + 1], g(E(G)) = [1, 8k + 1] and for each vertex a, b ∈ E(G) where a ∈ {xi, yi | i is odd} and b ∈

{xi, yi | i is even}, g(a) ≤ 4k − 1 and g(b) > 4k − 1. It follows that g is an α-labelling of KP(4k, 4k, 1).

Example 4.5. We give a metamorphosis of a KP(4, 4, 1)-design (taking n = 9 and x = 1) of order (2 × 9 × 4) + 1 = 73into a 4-cycle system of the same order.

Fig. 5 gives four starters for a KP(4, 4, 1)-design. Remove the dark edge from each KP(4, 4, 1) starter and rearrange themto give the starter shown in Fig. 6 for the metamorphosis.

M.W. Sutton / Discrete Mathematics 318 (2014) 71–77 77

Fig. 5. Four starters for KP(4, 4, 1) design of order 73.

Fig. 6. Single starter for the metamorphosis into a 4-cycle system.

5. Conclusion

Here the order v ≡ 1 (mod 2n(n − 1)) has been considered. Further orders v ≡ 0, n2, (n − 1)2 (mod 2n(n − 1)) couldalso be investigated for a metamorphosis of a G-design into a (G − e)-design, where the simple graphs G and G − e are notgiven explicitly. It would also be worthwhile trying to establish further results linking graph labellings and metamorphosesof G-designs into H-designs.

Acknowledgements

I would like to thank the Advanced Studies Program in Science (ASPinS) and the University of Queensland for makingthis research possible. Thanks also to an anonymous referee for comments.

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