Neighbor Sum Distinguishing Index of Sparse Graphs

Abstract

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 $${\chi^\prime_{\sum} }(G)$$ , we denote the smallest value k in such a coloring of G. Letmad(G) denote the maximum average degree of a graph G. In this paper, we prove that every normal graph with mad $$(G) < \tfrac{{10}}{3}$$ 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.