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

The neighbor-distinguishing total chromatic number $$\chi ''_{a}(G)$$ χ 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 $$\Delta =12$$ Δ = 12 , then $$13\le \chi ''_{a}(G)\le 14$$ 13 ≤ χ a ′ ′ ( G ) ≤ 14 , and moreover $$\chi ''_{a}(G)=14$$ χ a ′ ′ ( G ) = 14 if and only if G contains two adjacent 12-vertices.