arXiv subjects

Chunhui Ge

Publications and source records attributed to Chunhui Ge.

2 recordsLinked to original sources

arXiv2026

Edge-chromatic $4$-critical graphs and Overfull Conjecture for graphs with maximum degree $4$

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A graph $G$ is called edge-chromatic $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)< \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\Delta(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.

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↗