Neighbor sum distinguishing edge colorings of graphs with small maximum average degree

A proper edge-k-coloring of a graph G is an assignment of k colors 1, 2, · · · , k to the edges of G such that no two adjacent edges receive the same color. A neighbor sum distinguishing edge-k-coloring of G is a proper edge-k-coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident with u is different from the sum of colors taken on the edges incident with v. By ndi∑(G), we denote the smallest value k in such a coloring of G. The maximum average degree of G is mad(G) = max{2|E(H)|/|V (H)|}, where the maximum is taken over all the non-empty subgraphs H of G. In this paper, we obtain that if G is a graph without isolated edges and mad(G) < 8/3, then ndi∑(G) ≤ k where k = max{∆(G) + 1, 6}. It partially confirms the conjecture proposed by Flandrin et al.