On the Hilton-Zhao vertex-splitting conjecture

Abstract

Let $G$ be a simple graph with order $n$, maximum degree $\hat{I}”(G)$, and chromatic index $\ddot{I}{‡}^{\prime }(G)$, respectively. A graph $G$ is edge-chromatic critical if $\ddot{I}{‡}^{\prime }(H)<\ddot{I}{‡}^{\prime }(G)$ for every proper subgraph $H$ of $G$. Assume that $G$ is an $n$-vertex connected regular Class $1$ graph, and let $G^{\ast }$ be obtained from $G$ by splitting one vertex into two vertices. Hilton and Zhao in 1997 proposed the vertex-splitting conjecture: if $\hat{I}”(G)>\frac{n}{3}$, then $G^{\ast }$ is edge-chromatic critical. Recently, Cao, Chen, and Shan (Discrete Math. 2022) verified the conjecture for $\hat{I}”(G)\ge \frac{3n}{4}$. In this paper, we confirm the conjecture for $\hat{I}”(G)\ge \frac{2n-2}{3}$.