A new improvement to the Overfull Conjecture

TL;DR

This paper advances the Overfull Conjecture in graph theory by improving the bounds under which $ ext{D}$-critical graphs are guaranteed to be overfull, impacting algorithms for edge coloring.

Contribution

The paper improves the known bounds related to the Overfull Conjecture, extending the conditions under which the conjecture holds true.

Findings

Improved the bound from $ ext{D}(G)-7 ext{delta}(G)/4 extgreater (3n-17)/4$ to $ ext{D}(G)-5 ext{delta}(G)/3 extgreater (2n-7)/3$.

Confirmed the conjecture for a broader class of graphs.

Strengthened the theoretical foundation for polynomial-time algorithms in graph edge coloring.

Abstract

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 $\D(G)-5\delta(G)/3\ge (2n-7)/3$.