Strict neighbor-distinguishing edge coloring of planar graphs

Abstract

A proper edge coloring of a graph $G$ is strict neighbor-distinguishing if for any two adjacent vertices $u$ and $v$, the set of colors used on the edges incident to $u$ and the set of colors used on the edges incident to $v$ are not included with each other. The strict neighbor-distinguishing index of $G$ is the minimum number $\chi'_{\rm~snd}(G)$ of colors in a strict neighbor-distinguishing edge coloring of $G$. It is conjectured that every simple graph $G$ without leaves satisfies $\chi'_{\rm~snd}(G)\le~2\Delta$ except a special graph $H_{\Delta}$. The currently best-known upper bound is $\chi'_{\rm~snd}(G)\le~3\Delta-1$. An interesting question is: Which graphs $G$ without leaves satisfy $\chi'_{\rm~snd}(G)\le~\Delta+C$, where $C$ is a constant? This paper answers partially this 问题, i.e., it is proved that if $G$ is a planar graph with a girth of at least five, then $\chi'_{\rm~snd}(G)\le~\Delta+25.$