JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (6): 36-43, 70.doi: 10.6040/j.issn.1671-9352.0.2023.192

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Vertex-distinguishing E-total coloring of complete bipartite graph K12, n for 12≤n≤88

Kaiyang HU1(),Mingfang HUANG1,*(),Baolin MA2   

  1. 1. School of Science, Wuhan University of Technology, Wuhan 430070, Hubei, China
    2. School of Mathematics and Science, Henan Institute of Science and Technology, Xinxiang 453003, Henan, China
  • Received:2023-04-28 Online:2024-06-20 Published:2024-06-17
  • Contact: Mingfang HUANG E-mail:2776982452@qq.com;ds_hmf@126.com

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Abstract:

An E-total coloring of graph G is a mapping f : $V \cup E \rightarrow\{1, 2, \cdots, k\} $ such that for each edge $ e=u v \in E(G), f(e) \neq f(u)$, $ $ $ f(e) \neq f(v) \text { and } f(u) \neq f(v)$. For an E-total coloring f of a graph G, let C(v) denote the set of colors of vertex v and the edges incident with v. If $ C(u) \neq C(v) \text { where } u, v \in V(G) \text { and } u \neq v$, then that f is a k-vertex-distinguishing E-total coloring of graph G, or simply k-VDET coloring. This paper proves 6-VDET coloring and 7-VDET coloring of complete bipartite graph K12, n for 12≤n≤28 and 29≤n≤88, respectively.

Key words: complete bipartite graph, E-total coloring, vertex-distinguishing E-total coloring

CLC Number: 

  • O157.5

Table 1

Coloring method of vertex uiand its incident edges of K12, 28 when 12≤i≤28"

限制条件 顶点ui对应的集合 顶点ui及其关联边染色
1≤a≤3, 3≤b≤4, a≠b {a, b} a; b, b, b, b, b, b, b, b, b, b, b, b
3≤a≤4 {1, 2, a} 1;2, a, a, 2, 2, a, 2, 2, a, 2, a, a
1≤a≤2 {a, 3, 4} a; 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4
1≤a≤2, 3≤b≤4 {a, b, 5} a; b, b, b, b, b, b, 5, b, b, 5, b, b
1≤a≤2, 3≤b≤4 {a, b, 6} a; 6, b, b, b, b, b, b, b, 6, b, b, b
3≤a≤4 {1, 2, a, 5} 1;2, a, a, a, a, 5, a, a, a, 5, a, a
3≤a≤4 {1, 2, a, 6} 1;6, a, a, 2, a, a, a, a, 6, a, a, a
1≤a≤2, 3≤b≤4 {a, b, 5, 6} a; b, b, b, b, b, b, b, b, 6, 5, b, b
3≤a≤4 {1, 2, a, 5, 6} 1;a, a, a, a, a, a, a, a, 6, 5, a, a
{1, 2, 3, 4} 2;1, 1, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3

Table 2

Coloring method of vertex ui and its incident edges of K12, 88 when 12≤i≤88"

限制条件 顶点ui对应的集合 顶点ui及其关联边染色
1≤a≤4, 3≤b≤5, a≠b {a, b} a; b, b, b, b, b, b, b, b, b, b, b, b
3≤a≤5 {1, 2, a} 1;2, a, a, a, 2, 2, a, a, a, 2, a, a
1≤a≤3, 3≤b≤4, 4≤c≤5, a≠b≠c {a, b, c} a; b, c, c, c, c, c, c, c, c, c, c, c
1≤a≤2, 3≤b≤5 {a, b, 6} a; b, b, b, b, b, b, 6, b, b, b, 6, b
1≤a≤2, 3≤b≤5 {a, b, 7} a; b, b, b, b, b, b, b, b, b, 7, b, 7
3≤a≤4, 4≤b≤5, a≠b {a, b, 6} a; b, b, b, b, b, b, 6, b, b, b, 6, b
3≤a≤4, 4≤b≤5, a≠b {a, b, 7} a; b, b, b, b, b, b, b, b, b, 7, b, 7
3≤a≤5, b=4, 5, 7, a≠b {1, 2, a, b} 1;a, a, a, a, 2, 2, a, 2, 2, b, 2, 2
3≤a≤5 {1, 2, a, 6} 1;a, a, a, a, 2, 2, 6, 2, 2, 2, 6, 2
1≤a≤2, 3≤b≤4, 4≤c≤5, d=4, 5, 7, b≠c≠d {a, b, c, d} a; c, c, c, c, b, b, c, b, b, d, b, b
1≤a≤2, 3≤b≤4, 4≤c≤5, b≠c {a, b, c, 6} a; b, b, b, b, b, b, 6, b, b, b, 6, c
3≤a≤4, 4≤b≤5, c=5, 7, a≠b≠c {1, 2, a, b, c} 1;a, a, a, a, 2, 2, b, 2, 2, c, 2, 2
3≤a≤4, 4≤b≤5, a≠b {1, 2, a, b, 6} 1;a, a, a, a, 2, 2, 6, 2, 2, 2, 6, b
3≤a≤5 {1, 2, a, 6, 7} 1;a, a, a, a, 2, 2, 6, 2, 2, 7, 6, 7
1≤a≤2, 3≤b≤4, 4≤c≤5, b≠c {a, b, c, 6, 7} a; b, b, b, b, c, c, 6, c, c, 7, 6, 7
4≤a≤5 {1, 2, 3, a, 6, 7} 2;3, a, 1, 1, 3, a, 6, 1, 1, 1, 6, 7
{1, 2, 4, 5, 6, 7} 2;4, 5, 1, 1, 4, 5, 6, 1, 1, 7, 6, 1
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