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arXiv · 2605.18783

On the Hilton-Zhao vertex-splitting conjecture

Xuli Qi·Yanrui Feng
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Abstract

Let $G$ be a simple graph with order $n$, maximum degree $\Delta(G)$, and chromatic index $\chi'(G)$, respectively. A graph $G$ is edge-chromatic critical if $\chi'(H)<\chi'(G)$ 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 $\Delta(G)>\frac{n}{3}$, then $G^*$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $\Delta(G)\ge\frac{3n}{4}$. In this paper, we confirm the conjecture for $\Delta(G) \ge\frac{2n-2}{3}$.

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Xuli Qi, Yanrui Feng. 2026-05-06. On the Hilton-Zhao vertex-splitting conjecture. https://arxiv.org/abs/2605.18783

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