Total colorings of planar graphs without chordal 6-cycles

A total kk-coloring of a graph GG is a coloring of V(G)∪E(G)V(G)∪E(G) using kk colors such that no two adjacent or incident elements receive the same color. The total chromatic number of GG is the smallest integer kk such that GG has a total kk-coloring. In this paper, it is proved that if GG is a planar graph with maximum degree Δ≥7Δ≥7 and without chordal 66-cycles, then the total chromatic number of GG is Δ+1Δ+1.