图的染色理论是图论的一个重要研究领域,求解图的色数被认为是一个NP-hard问题。对简单连通图G(V,E),存在一个正整数k,使得映射f :V(G)∪ E(G)→{1,2,…,K},如果对∀uv∈E(G),有f(u)≠f(uv),f(v)≠f(uv),且C(u)≠C(v),则称f是图G的点边邻点可区别全染色(又称为邻点可区别VE-全染色),而χ_at^ve (G)=min{k|k－VE－AVDTC},称为G的点边邻点可区别边色数(又称为邻点可区别VE-全色数),其中色集合C(u)={f(u)}∪{f(uv)|uv∈E(G)}。本文构造了两类冠图C_m·S_n和C_m·P_n,研究了两类冠图C_m·S_n和C_m·P_n的点边邻点可区别全染色。根据C_m·S_n和C_m·P_n的结构性质,用穷染递推的方法,得到了它们的相应色数,给出一种染色方案。

Graph coloring is one of the chief topics in the graph research, the solution of the chromatic number of the graph is an NP-hard problem. 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(uv), f(v)≠f(uv), C(u)≠C(v), f can be called the vertex-edge-adjacent vertex distinguishing total coloring of G (adjacent vertex distinguishing VE-total coloring of G), χ_at^ve(G)=min{k|k－VE－AVDTC} would be called the vertex-edge-adjacent vertex distinguishing total chromatic number of G (adjacent vertex distinguishing VE-total chromatic number of G), where vertex-edge-adjacent vertex distinguishing total coloring of G C(u)={f(u)}∪{f(uv)|uv∈E(G)}. In this paper, two kinds of crown graphs C_m·S_n and C_m·P_n are designed, the vertex-edge-adjacent vertex distinguishing total coloring of C_m·S_n and C_m·P_n are studied. According to the properties of two kinds of crown graphs C_m·S_n and C_m·P_n, by using colors one by one and in recursion, the vertex-edge-adjacent vertex distinguishing total chromatic numbers of two kinds of crown graphs C_m·S_n and C_m·P_n are obtained.