路的字典积的邻和可区别边染色

图G的正常k]-边染色σ是指颜色集合为k]={1，2，…，k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和，即w_σ（x）=∑_e?x σ（e），并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常k]-边染色，最小的k值称为G的邻和可区别边色数，记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_nH]的邻和可区别边色数的精确值，其中H分别为正则第一类图、路、完全图的补图.

Neighbor sum distinguishing edge coloring of the lexicographic product of paths

A properk]-edge coloring σ of graph G is a k-proper-edge-coloring of graph G using colors ink]={1, 2, …, k}. Let w_σ(x) denote the sum of the colors of edges incident with x, i.e., w_σ(x)=∑_e?x σ(e), and w_σ(x) is called the weight of the vertex x with respect to σ. A neighbor sum distinguishing edge coloring σ of G is a properk]-edge coloring of G such that no pair adjacent vertices receive the same weights. The smallest value k for which G has such a coloring is called the neighbor sum distinguishing edge chromatic number of G and denoted by χ'_∑(G). We obtained the exact values of this parameter for the lexicographic product P_nH] of a path P_n and a connected simple graph H, where H is a Class 1 regular graph, a path, the complement of a complete graph, respectively.