Total coloring of planar graphs without 6-cycles

A totalk-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 χ″(G)χ″(G) is the smallest integer kk such that GG has a total kk-coloring. Let GG be a planar graph with maximum degree Δ(G)Δ(G) and without 6-cycles. In this paper, it is proved that χ″(G)=Δ(G)+1χ″(G)=Δ(G)+1 if Δ(G)≥5Δ(G)≥5 and GG contains no 4-cycles, or Δ(G)≥6Δ(G)≥6 and GG contains no 5-cycles.