Adjacent vertex distinguishing colorings by sum of sparse graphs

A neighbor sum distinguishing edge-k-coloring, or nsd-k-coloring for short, of a graph G is a proper edge coloring of G with elements from {1, 2,..., k} such that no pair of adjacent vertices meets the same sum of colors of G. The definition of this coloring makes sense for graphs containing no isolated edges (we call such graphs normal). Let mad(G) and Delta(G) be the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every normal graph with Delta(G) >= 5 and mad(G) < 3 admits an nsd-(Delta(G) + 2)-coloring. Our approach is based on the Combinatorial Nullstellensatz and the discharging method. (C) 2015 Elsevier B.V. All rights reserved.