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

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k], where [k]={1,2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge 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 average degree of a graph G is ∑v∈V(G)d(v)|V(G)|; we denote it by ad(G). The maximum average degree mad(G) of G is the maximum of average degrees of its subgraphs. In this paper, we show that, if G is a graph without isolated edges and mad(G)≤52, then ndi∑(G)≤k, where k=max{Δ(G)+1,6}. This partially confirms the conjecture proposed by Flandrin et al.