Total coloring of planar graphs without adjacent chordal 6-cycles

Abstract

A total coloring of a graph G is a coloring such that no two adjacent or incident elements receive the same color. In this field there is a famous conjecture, named Total Coloring Conjecture, saying that the the total chromatic number of each graph G is at most $$\Delta +2$$Δ+2. Let G be a planar graph with maximum degree $$\Delta \ge 7$$Δź7 and without adjacent chordal 6-cycles, that is, two cycles of length 6 with chord do not share common edges. In this paper, it is proved that the total chromatic number of G is $$\Delta +1$$Δ+1, which partly confirmed Total Coloring Conjecture.