Vertex-distinguishing edge coloring of graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866915669654634496 |
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| author | Gao, Yuping Shan, Songling Wang, Guanghui Zhou, Yiming |
| author_facet | Gao, Yuping Shan, Songling Wang, Guanghui Zhou, Yiming |
| contents | Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $φ$ 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 $φ(v)$ for the set of colors assigned to the edges incident with $v$. The coloring $φ$ is called \emph{vertex-distinguishing} if $φ(u) \ne φ(v)$ 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 $χ'_{vd}(G)$. In 1997, Burris and Schelp conjectured that for every graph $G$ with at most one isolated vertex and no isolated edge, $ k(G) \;\le\; χ'_{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 $Δ(G) \ge \sqrt{2|V(G)|} + 4 $ and $δ(G) \ge 5$. For graphs that do not satisfy these conditions, the best known general upper bound on $χ'_{vd}(G)$ remains $|V(G)| + 1$, established in 1999 by Bazgan, Harkat-Benhamdine, Li, and Woźniak. In this paper, we prove that $χ'_{vd}(G) \le \floor{5.5k(G)+6.5}$, which represents a substantial improvement over the bound $|V(G)| + 1$ whenever $k(G) = o(|V(G)|)$. We further show that $χ'_{vd}(G) \le k(G) + 3$, for all $d$-regular graphs $G$ with $d \ge \log_2 |V(G)|\geq 8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex-distinguishing edge coloring of graphs Gao, Yuping Shan, Songling Wang, Guanghui Zhou, Yiming Combinatorics Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $φ$ 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 $φ(v)$ for the set of colors assigned to the edges incident with $v$. The coloring $φ$ is called \emph{vertex-distinguishing} if $φ(u) \ne φ(v)$ 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 $χ'_{vd}(G)$. In 1997, Burris and Schelp conjectured that for every graph $G$ with at most one isolated vertex and no isolated edge, $ k(G) \;\le\; χ'_{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 $Δ(G) \ge \sqrt{2|V(G)|} + 4 $ and $δ(G) \ge 5$. For graphs that do not satisfy these conditions, the best known general upper bound on $χ'_{vd}(G)$ remains $|V(G)| + 1$, established in 1999 by Bazgan, Harkat-Benhamdine, Li, and Woźniak. In this paper, we prove that $χ'_{vd}(G) \le \floor{5.5k(G)+6.5}$, which represents a substantial improvement over the bound $|V(G)| + 1$ whenever $k(G) = o(|V(G)|)$. We further show that $χ'_{vd}(G) \le k(G) + 3$, for all $d$-regular graphs $G$ with $d \ge \log_2 |V(G)|\geq 8$. |
| title | Vertex-distinguishing edge coloring of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.10827 |