Total coloring of planar graphs without adjacent short cycles

Abstract

In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let $$G=(V,E)$$G=(V,E) be a graph. Total coloring is a coloring of $$V\cup {E}$$VźE such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with $$\varDelta \ge 8$$Δź8. We proved that if for every vertex $$v\in V$$vźV, there exists two integers $$i_v,j_v\in \{3,4,5,6,7\}$$iv,jvź{3,4,5,6,7} such that v is not incident with adjacent $$i_v$$iv-cycles and $$j_v$$jv-cycles, then the total chromatic number of graph G is $$\varDelta +1$$Δ+1.