关键词: 广义字典积, 字典积, 点可区别边染色, 点可区别边色数

Abstract: For the generalized lexicographic product G［hn］ of a graph G and a sequence of vertex disjoint graphs hn=(Hi)i∈｛0,1,…,n-1｝, if HiH for i=0,1,…,n-1, then G［hn］=G［H］, where G［H］ is the lexicographic product of two graphs. The minimum number of colors required for a vertexdistinguishing edge coloring of G is called the vertexdistinguishing edge chromatic number of G and denoted by  χ′vd(G). For any graph G with  χ′vd (G)=Δ(G), two upper bounds on the parameter χ′vd (G［hn］) are provided in this paper and those are proved to be attainable precisely, where every Hi is a simply graph of order m. It is also proved that if χ′vd(G)=Δ(G), χ′vd(H)=Δ(H) and Δ(G［H］)=Δ(H［G］), then χ′vd (G［H］)=χ′vd (H［G］), where G and H are simply graphs of order n and m, respectively.

Key words: vertex-distinguishing edge chromatic number, lexicographic product, generalized lexicographic product, vertexdistinguishing edge coloring