Neighbor sum distinguishing colorings of graphs with maximum average degree less than 3712

Let G be a graph and let its maximum degree and maximum average degree be denoted by .(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv . E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by ...(G). Flandrin et al. proposed the following conjecture that ... (G) . .(G) + 2 for any connected graph with at least 3 vertices and G . C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < 3712 and .(G) . 7.