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

For any vertex u epsilon V(G), let T (N) (u) = {u} boolean OR {uv vertical bar uv epsilon E(G), v epsilon v(G)} boolean OR {v epsilon v(G)vertical bar uv epsilon 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) vertical bar x epsilon T-N (u)}. For any two adjacent vertices x and y of V(G) such that C-f(x) not equal 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 chi(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 Delta(G) + 1 <= chi(ast) (G) <= Delta(G) + 2 for any tree or unique cycle graph G.