图的染色理论在模式识别、生物信息、社交网络和电力网络上有重要的应用。对于图G的一个点染色φ:V(G)→{ 1,2,⋯,k },若满足对任意非孤立点v∈V(G),都存在c∈{ 1,2,⋯,k }使得| φ−1(c)∩N(v) |是一个奇数,则称φ是图G的一个奇k-染色。特别地,若| φ−1(c)∩N(v) |=1,则称φ是图G的一个正常无冲突k-染色。图G的奇(正常无冲突)色数是使图G有一个奇(正常无冲突) k-染色的k的最小值,记作χo(G)(χpcf(G))。本文研究笛卡尔乘积图的奇染色和正常无冲突染色,确定了Pm和Pn的笛卡尔乘积图的奇色数和正常无冲突色数,确定了奇色数的上界,丰富了图的染色理论,为实践应用提供了理论指导。The coloring theory of graphs has important applications in pattern recognition, biological information, social networks and power networks. For a vertex coloring φ:V(G)→{ 1,2,⋯,k }of a graph G, it is called an odd k-coloring of G if for each non-isolated vertex v∈V(G), there exist c∈{ 1,2,⋯,k }such that | φ−1(c)∩N(v) |is odd. Especially, it is called a proper conflict-free k-coloring of G when | φ−1(c)∩N(v) |=1. The odd (proper conflict-free) chromatic number of a graph G, denoted by χo(G)(χpcf(G)), is the minimum k such that G has an odd (proper conflict-free) k-coloring. In this paper, we study the odd coloring and proper conflict-free coloring of cartesian product graph, determine the odd chromatic number and PCF chromatic number of cartesian product graph of Pmand Pn, and determine the upper bound of odd chromatic number, which enriches the coloring theory of graphs and provides theoretical guidance for practical application.
