On the adjacent-vertex-strongly-distinguishing total coloring of graphs

Abstract

For any vertex u ∊ V(G), let T _N (u) = {u} ∪ {uυ|uυ ∊ E(G), υ ∊ υ(G)} ∪ {υ ∊ υ(G)|uυ ∊ E(G) and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C _f(u) = {f(x) | x ∊ T _N (u)}. For any two adjacent vertices x and y of V(G) such that C _f(x) ≠ C _f(y), we refer to f as a k-avsdt-coloring of G (“avsdt” is the abbreviation of “ adjacent-vertex-strongly-distinguishing total”). The avsdt-coloring number of G, denoted by χ_ast(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We prove Δ(G) + 1 ⩽ χ_ast(G) ⩽ Δ(G) + 2 for any tree or unique cycle graph G.