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.

Key words: complete tripartite graph, IE-total coloring, vertex-distinguishing IE-total coloring, vertex-distinguishing IE-total chromatic number