| LetG(V,E) be a simple connected graph,k is a positive integer.If a mapping of f from (G) ∩ E(G) to{1,2,...,k} satisfies the condition (?) u,v ∈ V(G), uv ∈ E(G),f(u)≠f (v), f (v)≠f (uv),f (u)≠f(uv), C(u)≠C(v), C(u)={f(u)}∪{f(uv)|uv ∈ E(G),u,v ∈ V(G)}and ||Si|-|Sj||≤1(i≠j,1≤i,j≤k), where Si= Vi ∪ Ei={uv|f(uv)= i,uv ∈ E(G)},Vi={v| f(v)= i,v ∈ V(G)}, then f is called an adjacent vertex distinguishing equitable E-total coloring of graph G, which the required minimal number of k is called the adjacent vertex iistinguishing equitable E-total chromatic mumber of G.In this paper, the analysis of composite structure method, the distribution of the overall ;olor method and the exhaustive method are used to study the adjacent vertex distinguishing equitable E-total coloring question of graph Wn 2,Fn,2 and the complete bipartite graphs Km,n ind corona,direct product and k-square of some simple graphs. On the basis of the above, we obtain the corresponding chromatic numbers of them.The paper is divided into five parts:In the first part, we introduce some fundamental concepts and terminologists and ymbols that related to this paper.In the second part, we discuss the adjacent vertex distinguishing equitable E-total coloring question of several corona graphs, and the corresponding chromatic numbers of them were given.In the third part, we discuss the adjacent vertex distinguishing equitable E-total coloring question of formed double wheel and double fan graph by wheel Wn and fan Fn, and he complete bipartite graphs.In the forth part, we study the adjacent vertex distinguishing equitable E-total coloring question of three kinds of direct product graphs.In the last part, we mainly discuss the adjacent vertex distinguishing equitable E-total coloring question of k-square graphs Cn2, Pn2, Cn3, Pn3 and Cn(3),Pn(3) formed by Cn and Pn. |
