Neighbor Sum Distinguishing Chromatic Index of Sparse Graphs via the Combinatorial Nullstellensatz

Let Ф ： E（G）→ {1, 2,…, k}be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if ∑eЭu Ф（e）≠∑eЭu Ф（e） for each edge uv∈E（G）.The smallest value k for which G has such a coloring is denoted by χ＇Σ（G） which makes sense for graphs containing no isolated edge（we call such graphs normal）. It was conjectured by Flandrin et al. that χ＇Σ（G） ≤△（G） ＋ 2 for all normal graphs,except for C5. Let mad（G） = max{（2｜E（H）｜）/（｜V（H）｜）｜HЭG}be the maximum average degree of G. In this paper,we prove that if G is a normal graph with△（G）≥5 and mad（G） 〈 3-2/（△（G））, then χ＇Σ（G）≤△（G） ＋ 1. This improves the previous results and the bound △（G） ＋ 1 is sharp.