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a b s t r a c tthe sum of the color of a vertex v and the colors of all incident edges of, we denote the smallest value k in such a coloring of G. In this paper, we show that if G is a planar graph with ∆(G) ≥ 14, then χ ′′ nsd (G) ≤ ∆(G) + 2.

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A total [k]-coloring of a graph G is a mapping φ:denote the set of colors of the edges incident to v and the color of v. If for each edge uv, C φ (u) = C φ (v), we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. χ a (G) denotes the smallest value k in such a coloring of G. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph G with maximum degree ≥ 8 contains no adjacent 4-cycles, then χ a (G) ≤ + 3.