Neighbor Sum Distinguishing Index of Sparse Graphs

A proper k-edge coloring of a graph G is an assignment of one of k colors to each edge of G such that there are no two edges with the same color incident to a common vertex. Let f（v） denote the sum of colors of the edges incident to v. A k-neighbor sum distinguishing edge coloring of G is a proper k-edge coloring of G such that for each edge uv∈E（G）, f（u）≠f（v）. By χ'∑（G）, we denote the smallest value k in such a coloring of G. Let mad（G） denote the maximum average degree of a graph G. In this paper, we prove that every normal graph with mad（G） <■ and Δ（G） ≥ 8 admits a(Δ（G） + 2)-neighbor sum distinguishing edge coloring. Our approach is based on the Combinatorial Nullstellensatz and discharging method.