Vertex coloring edge weightings of some square graphs

Abstract

A k-edge-weighting w of a graph G is an assignment of integer weight, w(e) ? {1, 2, . . . , k}, to each edge e. A k-edge-weighting w induces a vertex coloring c by defining c(u) = P u?e w(e) for every u ? V (G), where u ? e denote that u is an end-vertex of e. A k-edge-weighting w of a graph G is a vertex coloring of G if the induced coloring c is proper, i.e., c(u) 6= c(v) for any edge uv ? E(G). In this paper, vertex coloring edge weighting of square of Cartesian product of paths is considered.