Neighbor-distinguishing total coloring of planar graphs with maximum degree twelve

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The neighbor-distinguishing total chromatic number $$\chi ''_{a}(G)$$ of a graph G is the minimum number of colors required for a proper total coloring of G such that any two adjacent vertices have different sets of colors. In this paper, we show that if G is a planar graph with $$\Delta =12$$, then $$13\le \chi ''_{a}(G)\le 14$$, and moreover $$\chi ''_{a}(G)=14$$ if and only if G contains two adjacent 12-vertices.