Journal of East China Normal University(Natural Science) ›› 2020, Vol. 2020 ›› Issue (6): 24-29.doi: 10.3969/j.issn.1000-5641.201911028

• Mathematics • Previous Articles     Next Articles

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

CHEN Xiang’en, YANG Weiguang   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
  • Received:2019-06-26 Published:2020-12-01

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

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