Towards the Overfull Conjecture

TL;DR

This paper advances the Overfull Conjecture in graph theory by proving it for large graphs with maximum degree close to the number of vertices, removing previous degree restrictions and confirming related conjectures.

Contribution

It proves the Overfull Conjecture for graphs with maximum degree at least (1 - ε)n, for small ε, without minimum degree constraints, extending prior results.

Findings

Proves the Overfull Conjecture for large graphs with high maximum degree.

Extends the validity of the conjecture beyond previous degree bounds.

Implications for polynomial-time algorithms and longstanding edge coloring conjectures.

Abstract

Let $G$ be a simple graph with maximum degree denoted as $\Delta(G)$. An overfull subgraph $H$ of $G$ is a subgraph satisfying the condition $|E(H)| > \Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture, stating that a graph $G$ with maximum degree $\Delta(G)> \frac{1}{3}|V(G)|$ has chromatic index equal to $\Delta(G)$ if and only if it does not contain any overfull subgraph. The Overfull Conjecture has many implications. For example, it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\Delta(G) > \frac{1}{3}|V(G)|$, and implies several longstanding conjectures in the area of graph edge colorings. In this paper, we make the first breakthrough towards the conjecture when not imposing a minimum degree condition on the graph: for any $0<\varepsilon \le \frac{1}{14}$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\ge n_0$ vertices with $\Delta(G) \ge (1-\varepsilon)n$, then the Overfull Conjecture holds for $G$. The previous best result in this direction, due to Chetwynd and Hilton from 1989, asserts the conjecture for graphs $G$ with $\Delta(G) \ge |V(G)|-3$. Our result also implies the Average Degree Conjecture of Vizing from 1968 for the same class of graphs $G$.