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

Abstract

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

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