Neighbor sum distinguishing index of planar graphs

Abstract

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors from [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 to u is different from the sum of colors taken on the edges incident to v. By nsdi(G), we denote the smallest value k in such a coloring of G. It was conjectured by Flandrin et al. that if G is a connected graph without isolated edges and G≠C5, then nsdi(G)≤Δ(G)+2. In this paper, we show that if G is a planar graph without isolated edges, then nsdi(G)≤max{Δ(G)+10,25}, which improves the previous bound (max{2Δ(G)+1,25}) due to Dong and Wang.