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2. On the edge-balance index sets of Chain Sums of K4-e
Yu-ge Zheng* Juan Lu* Sin-Min Lee**
*(Department of Mathematics, Henan Polytechinc University Jiaozuo ,P.R.China)
* (Department of Computer Science San Jose State Univer-
sity San Jose, CA95192,USA)
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3. On the edge-balance index sets of the first type of chain-sums ofⅠ
Introduction Ⅱ
On the edge-balance index sets of the first type of chain-sums of
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On the edge-balance index sets of the second type of chain-sums of
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4. Use the vertex and edge of graph labeling function
Introduction
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.
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5. Introduction
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 .
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6. Introduction 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.
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7. Introduction 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}.
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8. Main content
Definition 1. A labeling f of a graph G is said to be edge-friendly if
Definition 2. The edge-balance index set of the graph G, EBI(G), is defined as {| |: the edge labeling f is edge-friendly }
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9. Main content 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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10. Main content
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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11. Main content
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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12. Main content 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, …
, }.
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13. 一、 Main content 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}
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Lemma 2. The perfect index set of B(1) exists, and PEBI(B(1)) = {0, 1, 2}.
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14. Main content
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}.
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Lemma 5. The perfect index set of B(4) exists, and PEBI(B(4)) = {0, 1, 2, 3, 4, 5}.
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15. Main content
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}.
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Lemma 8. The perfect index set of B(7) exists, and PEBI(B(7)) = {0, 1, 2, … , 11}.
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16. Main content
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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17. Main content
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}.
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Lemma 14. The perfect index set of B(13) exists, and PEBI(B(12)) = {0, 1, 2, … , 20}.
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18. Main content
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.
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Theorem 4. The perfect index set of B(n) (n is odd) exist if and only if n = 1, 3, 5, 7, 9, 13.

19. type of chain-sums of K4-e
Main content 二、
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.
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Lemma 2 If m1≡ 1(mod 4) (m1= 4k+1 (k N)), then max{ EBI(H(n)) } = 6k-3n+2.
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20. Main content 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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21. Main content 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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22. Main content
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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23. Main content 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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24. 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),
[2] Ebrahim Salehi and Sin-Min Lee, Friendly index sets of trees, Congressus Numerantium 178 (2006), pp
[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)
[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)
[5] Harris Kwong, Sin-Min Lee and H.K. Ng ,On Friendly Index Sets of 2-regular graphs, Discrete Mathematics.,308 (2008)
[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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25. Thank you
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