Vertex-distinguishing edge coloring of graphs

TL;DR

This paper investigates the vertex-distinguishing edge coloring of graphs, providing new upper bounds on the minimum number of colors needed, which improve upon previous bounds especially for graphs with small lower bounds.

Contribution

The paper establishes a new upper bound of approximately 5.5 times the lower bound plus a constant for the vertex-distinguishing chromatic index, improving previous bounds for certain graph classes.

Findings

Proves $oldsymbol{oxed{ ext{vertex-distinguishing chromatic index} oldsymbol{oxed{ extstyle loor{5.5k(G)+6.5}}}}$ bounds.

Shows $oldsymbol{oxed{ ext{for } d ext{-regular graphs with } d ext{ large, } oldsymbol{oxed{ ext{index} ext{ } oldsymbol{ ext{bounded by } k(G)+3}}}}$.

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 $d \ge \log_2 |V(G)|\geq 8$.