A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number χa′′(G) of a graph G is the smallest integer k such that G can be totally colored using k colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph G with maximum degree 13 to have χa′′(G)=14 or χa′′(G)=15. Precisely, we show that if G is a planar graph of maximum degree 13, then 14≤χa′′(G)≤15; and χa′′(G)=15 if and only if G contains two adjacent 13-vertices.