Neighbor Sum Distinguishing Edge Colorings of Graphs with Small Maximum Average Degree

Abstract

A proper edge-k-coloring of a graph G is an assignment of k colors $$1,2,\ldots ,k$$ 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 $$uv\in E(G)$$ , 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 $${ {ndi}}_{\sum }(G)$$ , 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 $${ {ndi}}_{\sum }(G)\le k$$ where $$k=\max \{\Delta (G)+1,6\}$$ . It partially confirms the conjecture proposed by Flandrin et al. (Graphs Comb 29:1329–1336, 2013).