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 $$\phi $$ź: $$V(G)\cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$V(G)źE(G)ź[k]={1,2,ź,k} such that no two adjacent or incident elements in $$V(G)\cup E(G)$$V(G)źE(G) receive the same color. In a total [k]-coloring $$\phi $$ź of G, let $$C_{\phi }(v)$$Cź(v) denote the set of colors of the edges incident to v and the color of v. If for each edge uv, $$C_{\phi }(u)\ne C_{\phi }(v)$$Cź(u)źCź(v), we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. $$\chi ''_{a}(G)$$źaźź(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 $$\Delta \ge 8$$Δź8 contains no adjacent 4-cycles, then $$\chi ''_{a}(G)\le \Delta +3$$źaźź(G)≤Δ+3.