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On the edge-balance index sets of Chain Sums of K4-e

Yu-ge Zheng* Juan Lu* Sin-Min Lee**

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*(Department of Mathematics, Henan Polytechinc University Jiaozuo 454000,P.R.China)•* (Department of Computer Science San Jose State Univer-sity San Jose, CA95192,USA)

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Ⅰ Introduction

Ⅱ On the edge-balance index sets of the first type of chain-sums of

Ⅲ On the edge-balance index sets of the second type of chain-sums of eK 4

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Use the vertex and edge of graph labeling function

theory to study the graphs by B.M. Stewart introduced

in 1966, over the years, many domestic and foreign re-

searchers to work closely with the research in this area,

and accessed to a series of research results, such as gra-

phical construction method of edge-magic graphs and

edge-graceful graphs, their theoretical study and so on.

Introduction

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Boolean index sets of graphs are that make the vertex sets and the edge sets of graphs through the mapping function with corresponding, to study the charac-

teristics of various types of graphs and the inherent characteristics of graphs, Boolean index set theory can be applied to information engineering, commu-

unication networks、 computer science、 economic management、 medicine, etc. The edge-balance index set is an important issue in Boolean index set .

Introduction

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2Z

In this paper, we will introduce the edge-balance index

sets of chain sums of mainly. Let G be a graph

with vertex set V(G) and edge set E(G), and let Z2={0,

1}. A labeling f : E(G) → Z2 induces a partial vertex

labeling f+ : V(G) → Z2 defined by f+ (x) = 0 if the

edge labeling of f(xy) is 0 more than 1 and f+ (x) = 1

if the edge labeling of f(xy) is 1 more than 0.

Introduction

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f+ (x) is not defined if the number of edge labeling with

0 is equal to the number of edge labeling with 1. For i Z2, vf (i) = v (i) = card{v V(G) : f+ (V) = i} and

ef (i) = e (i) = card{e E(G) : f (e) = i}.

Introduction

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

Definition 2. The edge-balance index set of the graph G, EBI(G), is defined as {| |: the edge labeling f is edge-friendly }

)1()0( ff vv

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Definition 3. Let G be a graph and u, v be two distinct

vertices of G. We construct the graph Pn(G, {u, v}) as

follows. The vertex set V(Pn(G, {u, v})) is the union of

(n-1) copies of V(G). We denote the vertices u, v in the

copy of V(G) by ui, vi. The edge set E(Pn(G, {u, v})) is

the union of (n-1) copies of E(G) with vj and uj-1 ident-

ifined for j=1,2,…,n-2.We call Pn (G, {u, v}) the chain

sum of (G, {u, v}) by Pn.

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For n > 3, we denote Pn(K4-e, {x, z}) by B(n-1), B(n-1)

is said to be the first type of chain-sums of K4-e else; we

denote Pn(K4-e, {u, v}) by H(n-1), H(n-1) is said to be

the second type of chain-sums of K4-e. As shown in

Figure 1.

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Definition 4. Let G be a continuous subgraph of graph H(n) (or B(n)), and the number of 1-edge be m in G. If the difference between the number of 1-vertices and the number of 0-vertices is the largest in graph G, then the graph G is said to be the best m-edge graph.(We say an edge labeling of f(xy) is 1 to be 1-edge and an edge labeling of f(xy) is 0 to be 0-edge, an vertex labeling is 1 to be 1-vertex and an vertex labeling is 0 to be 0-vertex when the context is clear. )

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Definition 5. The complete index set of B(n), CEBI(B(n))is defined as EBI(B(n)) = {0, 1, … , k}.

Definition 6. The perfect index set of B(n), PEBI(B(n)), is defined as CEBI(B(n)) = {0+3i, 1+3i, 2+3i, i=0, 1, … , }.

2

1n

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On the edge-balance index sets of the first type of chain-sums of K4-e

Lemma 1. if EBI(B(2n-1)) = {0, 1, … , k}, then EBI (B(2n+1)) includes {0, 1, … , k}

Lemma 2. The perfect index set of B(1) exists, and PEBI(B(1)) = {0, 1, 2}.

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一、

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Lemma 3. The perfect index set of B(2) exists, and PEBI(B(2)) = {0, 1, 2}.

Lemma 4. The perfect index set of B(3) exists, and PEBI(B(3)) = {0, 1, 2, 3, 4, 5}.

Lemma 5. The perfect index set of B(4) exists, and PEBI(B(4)) = {0, 1, 2, 3, 4, 5}.

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Lemma 6. The perfect index set of B(5) exists, and PEBI(B(5)) = {0, 1, 2, 3,4, 5, 6, 7, 8}.

Lemma 7. The complete index set of B(6) exists but it’s perfect index set doesn’t exist, and CEBI(B(6)) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

Lemma 8. The perfect index set of B(7) exists, and PEBI(B(7)) = {0, 1, 2, … , 11}.

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Lemma 9. The perfect index set of B(8) exists, and PEBI(B(8)) = {0, 1, 2, … , 11}.

Lemma 10. The perfect index set of B(9) exists, and PEBI(B(8)) = {0, 1, 2, … , 14}.

Lemma 11. The perfect index set of B(10) exists, and PEBI(B(10)) = {0, 1, 2, … , 14}.

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Lemma 12. The complete index set of B(11) exists but it’s perfect index set doesn’t exist, and CEBI(B(11)) = {0, 1, 2, … , 16}.

Lemma 13. The perfect index set of B(12) exists, and PEBI(B(12)) = {0, 1, 2, … , 17}.

Lemma 14. The perfect index set of B(13) exists, and PEBI(B(12)) = {0, 1, 2, … , 20}.

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Theorem 1. The complete index set of B(n) all exist (n = 1, 2, … , 13).

Theorem 2. Except B(6) and B(11), the perfect index set of B(n) all exist (n = 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13).

Theorem 3. The complete index set of B(n) all exist.

Theorem 4. The perfect index set of B(n) (n is odd) exist if and only if n = 1, 3, 5, 7, 9, 13.

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On the edge-balance index sets of the second type of chain-sums of K4-e

Lemma 1 If m1 ≡ 0(mod 4) (m1 = 4k (k N)), then max{ EBI(H(n)) } = 6k-3n+1.

Lemma 2 If m1≡ 1(mod 4) (m1= 4k+1 (k N)), then max{ EBI(H(n)) } = 6k-3n+2.

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二、

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Lemma 3 If m1≡ 2(mod 4) (m1= 4k+2 (k N))), then max{ EBI( H(n) ) } = 6k-3n+3.

Lemma 4 If m1 ≡ 3(mod 4) (m1 = 4k+3 (k N)), then max{ EBI(H(n)) } = 6k-3n+5.

Theorem 1 If m1≡ 0(mod 4) (m1 = 4k (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+1}.

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Theorem 2 If m1≡ 1(mod 4) (m1= 4k+1 (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+2}.

Theorem 3 If m1≡ 2(mod 4) (m1= 4k+2 (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+3}.

Theorem 4 If m1 ≡ 3(mod4) (m1= 4k+3 (k N)), then EBI(H(n)) = {0, 1, 2, 3, …, 6k-3n+5}.

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Theorem 5 max{ EBI(H(n)) } = 3k+2 (k N) if and only if n = 4k+1 and n = 4k+2.

Theorem 6 max{ EBI(H(n)) } = 2(k+t)+4 if and only if n = 4k+3 (one to one correspondence between k and t, k = 0, 1, 2, 3, … , t = 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, … , in other words, t = 0 when k = 0, t = 0 when k = 1, followed by analogy).

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Theorem 7 max{ EBI(H(n)) } = 2(k+t)+3 if and only if n = 4k+4 (k = 0, 1, 2, 3, … , t = 0, 1, 1, 2, 2, 3, 3, 4,4, 5, 5, 6, 6, … , one to one correspondence between k and t)

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Reference [1] Bor-Liang Chen, Kuo-Ching Huang, Sin-Min Lee and Shi-Shan

Liu, On edge-balanced multigraphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 42(2002), 177-185 .

[2] Ebrahim Salehi and Sin-Min Lee, Friendly index sets of trees, Congressus Numerantium 178 (2006), pp. 173-183.

[3] Alexander Nien-Tsu Lee, Sin-Min Lee and Ho Kuen Ng, On The Balance Index Set of Graphs,

Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 135-150.

[4] Harris Kwong, Sin-Min Lee and D.G. Sarvate, On the Balance Index Sets of One-point Unions of Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 113-127.

[5] Harris Kwong, Sin-Min Lee and H.K. Ng ,On Friendly Index Sets of 2-regular graphs, Discrete Mathematics.,308 (2008) 5522-5532.

[6] Suh-Ryung Kim, Sin-Min Lee and Ho Kuen Ng, On Balancedness of Some Graph Constructions, Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 3-16.

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Thank you