Total coloring of planar graphs of maximum degree eight

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ″(G). It is known that if a planar graph G has maximum degree Δ⩾9, then χ″(G)=Δ+1. Recently Hou et al. (Graphs and Combinatorics 24 (2008) 91–100) proved that if G is a planar graph with maximum degree 8 and with either no 5-cycles or no 6-cycles, then χ″(G)=9. In this Note, we strengthen this result and prove that if G is a planar graph with maximum degree 8, and for each vertex x, there is an integer kx∈{3,4,5,6,7,8} such that there is no kx-cycle which contains x, then χ″(G)=9.