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On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill College (Funded by Stonehill Undergraduate Research Experience) 6th IWOGL 2010 University of Minnesota, Duluth October 22, 2010

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Page 1: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

On Edge-Balance Index Sets of L-Product of Cycles by Cycles

Daniel Bouchard, Stonehill College

Patrick Clark, Stonehill College

Hsin-hao Su, Stonehill College

(Funded by Stonehill Undergraduate Research Experience)

6th IWOGL 2010University of Minnesota, Duluth

October 22, 2010

Page 2: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Edge Labeling

A labeling f : E(G) Z2 induces a vertex partial labeling f+ : V(G) A defined by f+(x) = 0 if the edge labeling of f(x,y) is 0 more

than 1; f+(x) = 1 if the edge labeling of f(x,y) is 1 more

than 0; f+(x) is not defined if the number of edge

labeled by 0 is equal to the number of edge labeled by 1.

Page 3: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Example : nK2

EBI(nK2 ) is {0} if n is even and {2}if n is odd.

Page 4: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Definition of Edge-balance

Definition: A labeling f of a graph G is said to be edge-friendly if | ef(0) ef(1) | 1.

Definition: The edge-balance index set of the graph G, EBI(G), is defined as

{|vf(0) – vf(1)| : the edge labeling f is edge-friendly.}

Page 5: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Examples

Page 6: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Example : Pn

Lee, Tao and Lo[1] showed that

evenisnandn

oddisnandn

n

n

n

PEBI n

6}2,1,0{

5}1,0{

4}2,1{

3}0{

2}2{

)(

[1] S-M. Lee, S.P.B. Lo, M.F. Tao, On Edge-Balance Index Sets of Some Trees, manuscript.

Page 7: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Wheels

The wheel graph Wn = N1 +Cn-1 where V(Wn) = {c0} {c1,…,cn-1} and E(Wn) = {(c0,ci): i = 1, …, n-1} E(Cn-1).

W5W6

Page 8: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Edge Balance Index Set of Wheels

Chopra, Lee ans Su[2] proved: Theorem: If n is even, then

EBI(Wn) ={0, 2, …, 2i, …, n-2}. Theorem: If n is odd, then

EBI(Wn) = {1, 3, …, 2i+1, …, n-2}

{0, 1, 2, …, (n-1)/2}.

[2] D. Chopra, S-M. Lee, H-H. Su, On Edge-Balance Index Sets of Wheels, International Journal of Contemporary Mathematical Sciences 5 (2010), no. 53, 2605-2620.

Page 9: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

EBI(W6) = {0,2,4}

|v(0)-v(1)|= 0 |v(0)-v(1)|= 2 |v(0)-v(1)|= 4

Page 10: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

EBI(W5) = {0,1,2,3}

|v(0)-v(1)|= 0 |v(0)-v(1)|= 1 |v(0)-v(1)|= 2 |v(0)-v(1)|= 3

Page 11: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

A Lot of Numbers are Missing

EBI(W7) ={0, 1, 2, 3, 5}.

EBI(W9) ={0, 1, 2, 3, 4, 5, 7}.

EBI(W11) ={0, 1, 2, 3, 4, 5, 7, 9}.

EBI(W13) ={0, 1, 2, 3, 4, 5, 6, 7, 9, 11}.

EBI(W15) ={0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13}.

EBI(W17) ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15}.

Page 12: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

L-Product

Let H be a connected graph with a distinguished vertex s.

Construct a new graph G ×L (H,s) as follows: Take |V(G)| copies of (H,s), and identify each vertex of G with s of a single copy of H. We call the resulting graph the L-product of G and (H,s).

Page 13: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

L-Product Example

15 StC L 25 StC L

46 StC L

Page 14: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Generalized L-Product

More generally, the n copies of graphs to be identified with the vertices of G need not be identical.

Page 15: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Generalized L-Product

Let Gph* be the family of pairs (H,s), where H is a connected graph with a distinguished vertex s. For any graph G and any mapping : V(G) Gph* we construct the generalized L-product of G and , denoted G ×L , by identifying each v V(G) with s of the respective (v).

Page 16: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

L-Product of Cycles by Cycles

35 CC L45 CC L

Page 17: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Notations

Let f be an edge labeling of a cycle Cn.

We denote the number of edges of Cn which are labeled by 0 and 1 by f+ by eC(0) and eC(1), respectively.

We denote the number of vertices on Cn which are labeled by 0, 1, and not labeled by the restricted f+ by vC(0), vC(1), and vC(x), respectively

Page 18: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proposition

(Chopra, Lee and Su[2]) In a cycle Cn with a labeling f (not necessary edge friendly), assume that eC(0) > eC(1) > 1 and vC(x) = 2k > 0. Then we have

vC(1) = eC(1) - k.

and

vC(0) = n - eC(1) - k. [2] D. Chopra, S-M. Lee, H-H. Su, On Edge-Balance Index Sets of Wheels, International Journal of Contemporary Mathematical Sciences 5 (2010), no. 53, 2605-2620.

Page 19: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

EBI of Cycles

Lemma: For an edge labeling f (not necessary edge friendly) of a finite disjoint union of cycles , we have

Note that this EBI of the disjoint union of cycles depends on the number of 1-edges only, not how you label them.

i

ini

C

.1210 Ci

iCC envv

Page 20: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Maximal Edge-balance Index

Theorem: The highest edge-balance index of when m ≥ 5 is n if m is odd or n is even; n+1 if n is odd and m is even.

mn CC

Page 21: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof Idea

By the previous lemma, to maximize EBI, eC(1) has to be as small as it can be.

Thus, if we label all edges in Cn 1, it gives us the best chance to find the maximal EBI.

Thus, might yield the maximal EBI.

neeC 11

Page 22: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Less 1 inside, Higher EBI

.12

10

Ci

i

CC

en

vv

Page 23: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof

The number of edges of is

If n is even or m is odd, then

If n is odd and m is even, then

(Note that w.l.o.g we assume that .)

mn CC .1 mnmnn

.

2

11

mne

.

2

111

mne

10 ee

Page 24: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof (continued)

Since the outer cycles of contain all vertices, the EBI calculated by the previous lemma could be our highest EBI.

We already label all edges in Cn by 1. Thus, to not alter the label of the vertex adjuncts to a outer cycle, we have to have all two edges of outer cycle labeled by 1 too.

mn CC

Page 25: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Degree 4 Vertices

Page 26: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof (continued)

The above labeling requires n 1-edges for Cn and 2n 1-edges for outer cycles.

In order to have at least 3n 1-edges, the number of edges of must be greater or equal to 6n.

Thus, implies m must be greater or equal to 5.

mn CC

nmnmnn 61

Page 27: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Keep Degree 4 Unchanged

According to the formula

we can label the rest in any way without changing EBI.

,12

10

Ci

i

CC

en

vv

Page 28: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof (continued)

The highest EBI of is If n is even or m is odd, then

If n is odd and m is even, then

mn CC

.

2

1210 nn

mnnmvv

.1

2

11210

nn

mnnmvv

Page 29: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Switching Edges

By switching a 0-edge with an 1-edge adjacent to the inner cycle, we reduces the EBI by 1.

Page 30: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Switching Edges

By switching a 0-edge with an 1-edge adjacent to the inner cycle, we reduces the EBI by 1.

Page 31: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Main Results

Theorem: EBI( ) when m ≥ 5 is {0,1,2,…, n} if m is odd or n is even; {0,1,2,…, n+1} if n is odd and m is even.

mn CC

Page 32: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof

While creating an edge-labeling to yield the highest EBI, we label all edges adjacent to the inner cycle vertex 1.

Since the formula in the lemma says that the EBI of all outer cycles depends only on the number of 1-edges, we can label the edges adjacent to the edges adjacent the inner cycle vertex 0 without alter the EBI.

Page 33: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Special Edge-labeling to Yield the Highest EBI

According to the formula

we can label the rest in any way without changing EBI.

,12

10

Ci

i

CC

en

vv

Page 34: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof (continued)

Each outer cycle can reduce the EBI by 1 by switching edges.

Since there are n outer cycles, we can reduce the EBI by 1 n times.

Therefore, we have the EBI set contains {0,1,2,…, n} if m is odd or n is even; {1,2,…, n+1} if n is odd and m is even.

Page 35: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

Proof (continued)

When n is odd and m is even, a special labeling like the one on the right produces an EBI 0.

Page 36: On Edge-Balance Index Sets of L-Product of Cycles by Cycles Daniel Bouchard, Stonehill College Patrick Clark, Stonehill College Hsin-hao Su, Stonehill

When m = 3 or 4

Theorem: EBI( ) is {0, 1, 2, …, } if n is even.

{0, 1, 2, …, } if n is odd.

Theorem: EBI( ) is

{0, 1, 2, …, }.

4CCn

4

3n

3CCn

2n

4

13n