Journal of East China Normal University(Natural Science) >

2020 , Vol. 2020 >Issue 6: 24 - 29

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911028

Mathematics

Vertex-distinguishing E-total coloring of a complete bipartite graph K9, n (93 ≤ n ≤ 216)

  • CHEN Xiang’en ,
  • YANG Weiguang
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  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received date: 2019-06-26

  Online published: 2020-12-01

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Abstract

Let $G$ be a simple graph. A total coloring $f$ of $G$ is called an E-total coloring if no two adjacent vertices of $G$ receive the same color, and no edge of $G$ receives the same color as one of its endpoints. For an E-total coloring $f$ of a graph $G$, if $C(u)\neq C(v)$ for any two distinct vertices $u$ and $v$ of $V(G)$, where $C(x)$ denotes the set of colors of vertex $x$ and of the edges incident with $x$ under $f$, then $f$ is called a vertex-distinguishing E-total coloring of $G$. Let $\chi _{vt}^{e}(G)=\min\{k: G$ has a $k$-VDET coloring$\}.$ Then, $\chi _{vt}^{e}(G)$ is called the VDET chromatic number of $G$. By using contradiction, the method of a combinatorial analysis and the method of constructing specific coloring, the VDET coloring of a complete bipartite graph $K_{9, n}$ is discussed and the VDET chromatic number of $K_{9, n}\; (93\leqslant n\leqslant 216)$ is determined.

Key words： complete bipartite graph; E-total coloring; vertex-distinguishing E-total coloring; vertex-distinguishing E-total chromatic number

Cite this article

CHEN Xiang’en , YANG Weiguang . Vertex-distinguishing E-total coloring of a complete bipartite graph K9, n (93 ≤ n ≤ 216)[J]. Journal of East China Normal University(Natural Science), 2020 , 2020(6) : 24 -29 . DOI: 10.3969/j.issn.1000-5641.201911028

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