Adjacent-vertex-distinguishing Total Colorings Of Several Classes Of Graphs

Let G(V, E) be a connect graph with order at least2, k be a positive integer, and f be a mapping from V(G)∪E(G) to {1,2,…, k}. For any x∈V(G), let C(x)={f(x)}∪{f(xv)|xv∈E(G),v∈V(G)}. The set C(x) is called the color set of x. Let C(x)={1,2,…, k}-C(x). Proper k-total coloring f is called a k-adjacent vertex-distinguishing total coloring of graph G (in brief k-AVDTC) if for any uv∈E(G), we have C(u)≠C(v). The number χat(G)=min{k|G has a k-AVDTC} is called the adjacent vertex-distinguishing total chromatic number of graph G. The adjacent vertex distinguishing total chromatic number of graphs K3∨Kt, K4∨Kt, K5∨Kt C5∨Kt, C6∨Kt and C6∨Kt are determined in this paper.