Neighbor sum distinguishing edge colorings of sparse graphs

We consider proper edge colorings of a graph GG using colors of the set {1,…,k}{1,…,k}. Such a coloring is called neighbor sum distinguishing if for any uv∈E(G)uv∈E(G), the sum of colors of the edges incident to uu is different from the sum of the colors of the edges incident to vv. The smallest value of kk in such a coloring of GG is denoted by ndiΣ(G). Let mad(G) and Δ(G)Δ(G) denote the maximum average degree and the maximum degree of a graph GG, respectively. In this paper we show that, for a graph GG without isolated edges, if mad(G)<83, then ndiΣ(G)≤max{Δ(G)+1,7}; and if mad(G)<3, then ndiΣ(G)≤max{Δ(G)+2,7}.