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 $$V\cup {E}$$ V ? E such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of $$G$$ G is the minimum number of disjoint vertex independent sets covering a total graph of $$G$$ G . Here, let $$G$$ G be a planar graph with $$\varDelta \ge 8$$ Δ ? 8 . We proved that if for every vertex $$v\in V$$ v ? V , there exists two integers $$i_{v},j_{v} \in \{3,4,5,6,7,8\}$$ i v , j v ? { 3 , 4 , 5 , 6 , 7 , 8 } such that $$v$$ v is not incident with intersecting $$i_v$$ i v -cycles and $$j_v$$ j v -cycles, then the vertex chromatic number of total graph of $$G$$ G is $$\varDelta +1$$ Δ + 1 , i.e., the total chromatic number of $$G$$ G is $$\varDelta +1$$ Δ + 1 .