The overfull conjecture on graphs of odd order and large minimum degree

TL;DR

This paper proves the overfull conjecture for large odd-order graphs with high minimum degree, extending previous results that were limited to even-order graphs, thus advancing understanding of graph edge-coloring.

Contribution

It establishes the first proof of the overfull conjecture for odd-order graphs with large minimum degree, filling a significant gap in graph theory.

Findings

The conjecture holds for large odd-order graphs with minimum degree exceeding half the vertices.

The result confirms the conjecture for graphs with odd number of vertices and high minimum degree.

This work extends the validity of the overfull conjecture beyond even-order graphs.

Abstract

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and $G$ be a large graph on $n$ vertices with minimum degree at least $\frac{1}{2}(1+\varepsilon)n$. It was shown that the conjecture holds for $G$ if $n$ is even. In this paper, the same result is proved if $n$ is odd. As far as we know, this is the first result on the conjecture for graphs of odd order and with a minimum degree constraint.