The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

A total [k]-coloring of a graph G is a mapping $$\phi $$ ϕ : $$V(G)\cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$ V ( G ) ∪ E ( G ) → [ k ] = { 1 , 2 , … , k } such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) receive the same color. In a total [k]-coloring $$\phi $$ ϕ of G, let $$C_{\phi }(v)$$ C ϕ ( v ) denote the set of colors of the edges incident to v and the color of v. If for each edge uv, $$C_{\phi }(u)\ne C_{\phi }(v)$$ C ϕ ( u ) ≠ C ϕ ( v ) , we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. $$\chi ''_{a}(G)$$ 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 $$\Delta \ge 8$$ Δ ≥ 8 contains no adjacent 4-cycles, then $$\chi ''_{a}(G)\le \Delta +3$$ χ a ′ ′ ( G ) ≤ Δ + 3 .