On the Hilton-Zhao vertex-splitting conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917509391712256 |
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| author | Qi, Xuli Feng, Yanrui |
| author_facet | Qi, Xuli Feng, Yanrui |
| contents | Let $G$ be a simple graph with order $n$, maximum degree $Δ(G)$, and chromatic index $χ'(G)$, respectively. A graph $G$ is edge-chromatic critical if $χ'(H)<χ'(G)$ for every proper subgraph $H$ of $G$. Assume that $G$ is an $n$-vertex connected regular Class $1$ graph, and let $G^*$ be obtained from $G$ by splitting one vertex into two vertices. Hilton and Zhao in 1997 proposed the vertex-splitting conjecture: if $Δ(G)>\frac{n}{3}$, then $G^*$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $Δ(G)\ge\frac{3n}{4}$. In this paper, we confirm the conjecture for $Δ(G) \ge\frac{2n-2}{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_18783 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Hilton-Zhao vertex-splitting conjecture Qi, Xuli Feng, Yanrui Combinatorics Let $G$ be a simple graph with order $n$, maximum degree $Δ(G)$, and chromatic index $χ'(G)$, respectively. A graph $G$ is edge-chromatic critical if $χ'(H)<χ'(G)$ for every proper subgraph $H$ of $G$. Assume that $G$ is an $n$-vertex connected regular Class $1$ graph, and let $G^*$ be obtained from $G$ by splitting one vertex into two vertices. Hilton and Zhao in 1997 proposed the vertex-splitting conjecture: if $Δ(G)>\frac{n}{3}$, then $G^*$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $Δ(G)\ge\frac{3n}{4}$. In this paper, we confirm the conjecture for $Δ(G) \ge\frac{2n-2}{3}$. |
| title | On the Hilton-Zhao vertex-splitting conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.18783 |