Some bounds on the neighbor-distinguishing index of graphs

Abstract

A proper edge coloring of a graph $G$ is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor-distinguishing index of $G$ is the minimum number ${\chi }_a^{\prime }\left(G\right)$ of colors in a neighbor-distinguishing edge coloring of $G$.

Let $G$ be a graph with maximum degree $\Delta$ and without isolated edges. In this paper, we prove that ${\chi }_a^{\prime }\left(G\right)\le 2\Delta$ if $4\le \Delta \le 5$, and ${\chi }_a^{\prime }\left(G\right)\le 2.5\Delta$ if $\Delta \ge 6$. This improves a result in Zhang et al. (2014), which states that ${\chi }_a^{\prime }\left(G\right)\le 2.5\Delta +5$ for any graph $G$ without isolated edges. Moreover, we prove that if $G$ is a semi-regular graph (i.e., each edge of $G$ is incident to at least one $\Delta$-vertex), then ${\chi }_a^{\prime }\left(G\right)\le \frac{5}{3}\Delta +\frac{13}{3}$.