• Mathematics •

Vertex-distinguishing IE-total coloring of ${K_{5,\;5,\;p}}$ $(p \geqslant 2\;028)$

Ruimin YAN(), Xiang’en CHEN*()

1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China
• Received:2020-11-27 Online:2022-03-25 Published:2022-03-28
• Contact: Xiang’en CHEN E-mail:YanruiM@163.com;chenxe@nwnu.edu.cn

Abstract:

Let $G$ be a simple graph. A total coloring $f$ of $G$ is called an IE-total coloring if $f(u)\neq f(v)$ for any two adjacent vertices $u$ and $v$ , where $V(G)$ denotes the set of vertices of $G$ . For an IE-total coloring $f$ of $G$ , the set of colors $C(x)$ (non-multiple sets) of vertex $x$ under $f$ of $G$ is the set of colors of vertex $x$ and of the edges incident with $x$ . If any two distinct vertices of $G$ have distinct color sets, then $f$ is called a vertex-distinguishing IE-total coloring of $G$ . We explore the vertex distinguishing IE-total coloring of complete tripartite graphs $K_{5,5,p}$ $(p \geqslant 2\;028)$ through the use of multiple methods, including distributing the color sets in advance, constructing the colorings, and contradiction. The vertex-distinguishing IE-total chromatic number of $K_{5,5,p}$ $(p \geqslant 2\;028)$ is determined.

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