项链的若干染色问题

 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。

Several Coloring Problems Involving the Necklace

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.