Adjacent vertex distinguishing colorings by sum of sparse graphs

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