A sufficient condition for planar graphs with maximum degree 8 to be 9-totally colorable

A total k-coloring of a graph G is a coloring of V (G) . E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number ..(G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree . . 9, then ..(G) = .+1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then ..(G) = 9.