Yanrui Feng | Searcharxiv

arXiv2026

On the Hilton-Zhao vertex-splitting conjecture

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 $\Delta(G) \ge\frac{2n-2}{3}$.

math.CO↗

arXiv2025

A new improvement to the Overfull Conjecture

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\delta(G)$ and chromatic index $\chi'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $\chi'(G)=\D(G)+1$ and $\chi'(H)\textless \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor n/2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\D$-critical graph $G$ with $\D(G)\textgreater\frac{n}{3}$ is overfull. The Overfull Conjecture has many implications, such as that it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\D(G)\textgreater\frac{n}{3}$, and implies several longstanding conjectures in the area of graph edge coloring. Recently, Cao, Chen, Jing and Shan (SIAM J. Discrete Math. 2022) verified the Overfull Conjecture for $\D(G)-7\delta(G)/4\ge (3n-17)/4$. In this paper, we improve it for $\D(G)-5\delta(G)/3\ge (2n-7)/3$.

math.CO↗