图的染色问题是图论研究的经典领域,在网络结构和实际生活中都有着广泛的应用。染色问题是近年来图论研究的热点,全染色,特别是邻点可区别全染色又是染色问题中的难点。本文研究了当h≥3 (h能确定项链的顶点个数,N_h中的h表示项链有2h+2个顶点)时,项链的邻点可区别全染色、点边邻点可区别全染色和关联邻点可区别全染色。通过在项链的点边集合与色集合之间构造一种一一对应关系,得到它们的色数分别是5、3、4,同时给出了具体的染色方案。

The coloring problem of graph is the classical field of graph theory which is widely used in the network structure and practical life. The coloring problem is becoming a hot topic in recent years. However, the total coloring, especially adjacent vertex-distinguishing total coloring is a difficult point of the coloring problem. For a necklace, the adjacent vertex-distinguishing total coloring, the adjacent vertex-distinguishing vertex edge total coloring, and the incidence adjacent vertex-distinguishing total coloring are discussed when h≥3 (h is able to determine the number of vertices of necklace, h means that the necklace has 2h+2 vertices in the N_h). Through setting up a corresponding relation between the set of vertices and edges and the set of color, the corresponding chromatic numbers of the adjacent vertex-distinguishing total coloring, the adjacent vertex-distinguishing vertex edge total coloring, and the incidence adjacent vertex-distinguishing total coloring are obtained, the chromatic numbers for a necklace are five, three, and four, respectively. At the same time, the corresponding coloring schemes are given.