Edge-partitions of graphs and their neighbor-distinguishing index

A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by χa′(G). A graph is normal if it contains no isolated edges. Let G be a normal graph, and let Δ(G) and χ′(G) denote the maximum degree and the chromatic index of G, respectively. We modify the previously known techniques of edge-partitioning to prove that χa′(G)≤2χ′(G), which implies that χa′(G)≤2Δ(G)+2. This improves the result in Wang et al. (2015), which states that χa′(G)≤52Δ(G) for any normal graph. We also prove that χa′(G)≤2Δ(G) when Δ(G)=2k, k is an integer with k≥2.