Incidence Adjacent Vertex-Distinguishing Total Coloring of Graphs

Let G(V, E) be a simple graph, k is a positive integer, f is a mapping from V(G) ¿ E(G) to {1, 2, ..., k} such that ¿uv ¿ E(G), then f(u) ¿ f(v); ¿uv,vw ¿ E(G), u ¿ w,f(uv) ¿ f(vw); ¿uv ¿ E(G),C(u) ¿ C(v), we say that f is the incidence-adjacent vertex distinguishing total coloring of G. The minimum number of k is called the incidence-adjacent vertex distinguishing total chromatic number of G. Where C(u) = {f(u)}¿{f(uv)|uv ¿ E(G)}. In this paper, we discuss some graphs whose incidence-adjacent vertex distinguishing total chromatic number is just ¿, ¿ + 1, ¿ + 2, and present a conjecture that the incidence-adjacent vertex distinguishing total chromatic number of a graph is no more than ¿ + 2.