Total Coloring of Planar Graphs Without Some Chordal 6-cycles

Abstract

A $$k$$ -total-coloring of a graph $$G$$ is a coloring of vertex set and edge set using $$k$$ colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if $$G$$ is a planar graph with maximum $$\Delta \ge 8$$ and every 6-cycle of $$G$$ contains at most one chord or any chordal 6-cycles are not adjacent, then $$G$$ has a $$(\Delta +1)$$ -total-coloring.