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

Abstract The neighbor-distinguishing total chromatic number χ a ′ ′ ( 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 χ a ′ ′ ( G ) = 14 or χ a ′ ′ ( G ) = 15 . Precisely, we show that if G is a planar graph of maximum degree 13, then 14 ≤ χ a ′ ′ ( G ) ≤ 15 ; and χ a ′ ′ ( G ) = 15 if and only if G contains two adjacent 13-vertices.

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