Vertex-distinguishing edge coloring of graphs

Abstract

Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $\varphi$ of $G$ is an assignment of colors from $\{1,\ldots,k\}$ to the edges of $G$ such that no two adjacent edges receive the same color. For a vertex $v \in V(G)$, we write $\varphi(v)$ for the set of colors assigned to the edges incident with $v$. The coloring $\varphi$ is called \emph{vertex-distinguishing} if $\varphi(u) \ne \varphi(v)$ for every pair of distinct vertices $u,v \in V(G)$. A vertex-distinguishing edge-$k$-coloring exists if and only if $G$ has at most one isolated vertex and no isolated edge. The least integer $k$ for which such a coloring exists is called the \emph{vertex-distinguishing chromatic index} of $G$, denoted $\chi'_{vd}(G)$. In 1997, Burris and Schelp conjectured that for every graph $G$ with at most one isolated vertex and no isolated edge, $ k(G) \;\le\; \chi'_{vd}(G) \;\le\; k(G)+1$, where $k(G)$ is the natural lower bound required for a vertex-distinguishing coloring in $G$. In 2004, Balister, Kostochka, Li, and Schelp verified the conjecture for graphs $G$ satisfying $\Delta(G) \ge \sqrt{2|V(G)|} + 4 $ and $\delta(G) \ge 5$. For graphs that do not satisfy these conditions, the best known general upper bound on $\chi'_{vd}(G)$ remains $|V(G)| + 1$, established in 1999 by Bazgan, Harkat-Benhamdine, Li, and Wo\'zniak. In this paper, we prove that $\chi'_{vd}(G) \le \floor{5.5k(G)+6.5}$, which represents a substantial improvement over the bound $|V(G)| + 1$ whenever $k(G) = o(|V(G)|)$. We further show that $\chi'_{vd}(G) \le k(G) + 3$, for all $d$-regular graphs $G$ with $d \ge \log_2 |V(G)|\geq 8$.