路的联的邻和可区别边染色

Neighbor sum distinguishing edge coloring of the join of paths

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wσ(x)表示顶点x关联边的颜色之和,即wσ(x)=∑x∈eσ(e),并称wσ(x)为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)。本文给出了两条不同阶路的联的邻和可区别边色数的精确值。另外,得到了同阶路的邻和可区别边色数的上界。

A proper[k]-edge coloringσof a graph G is a k-proper-edge-coloring of G using colors in[k]={1,2,…,k},let wσ(x)denote the sum of the colors of edges incident with x,i.e.,wσ(x)=∑x∈eσ(e),and wσ(x)is called the weight of the vertex x with respect toσ.A neighbor sum distinguishing edge coloringσof G is a proper[k]-edge coloring of G such that no pair adjacent vertices receive the same weight.The smallest value k for which G has such a coloring is called the neighbor sum distinguishing edge chromatic number of G and denoted byχ′∑(G).The exact values of the neighbor sum distinguishing edge chromatic number of the join of two paths with different orders are given.The upper bound of the neighbor sum distinguishing edge chromatic number of the join of two paths with same orders is obtained.