Minimum total coloring of planar graph

Abstract

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of $V\cup {E}$ such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of $G$ is the minimum number of disjoint vertex independent sets covering a total graph of $G$. Here, let $G$ be a planar graph with $\varDelta \ge 8$. We proved that if for every vertex $v\in V$, there exists two integers $i_{v},j_{v} \in \{3,4,5,6,7,8\}$ such that $v$ is not incident with intersecting $i_v$-cycles and $j_v$-cycles, then the vertex chromatic number of total graph of $G$ is $\varDelta +1$, i.e., the total chromatic number of $G$ is $\varDelta +1$.