On total 9-coloring planar graphs of maximum degree seven

Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Delta(G) is the maximum degree of G, then no graph has a total Delta-coloring, but Vizing conjectured that every graph has a total (Delta + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Delta less than or equal to 7 has a total 9-coloring by means of the discharging method, (C) 1999 John Wiley & Sons, Inc.