Edge coloring graphs with large minimum degree

TL;DR

This paper proves that for large graphs with high minimum degree, the chromatic index equals the maximum degree if and only if the graph contains no overfull subgraph, supporting a long-standing conjecture in graph theory.

Contribution

The paper establishes an asymptotic characterization of edge-coloring in graphs with large minimum degree, confirming the overfull conjecture for such graphs.

Findings

Graphs with minimum degree at least (1+ε)n have chromatic index equal to maximum degree if and only if no overfull subgraph exists.

Supports the overfull conjecture in an asymptotic setting for large graphs.

Generalizes the 1-factorization conjecture for graphs with high minimum degree.

Abstract

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. The 1-factorization conjecture is a special case of this overfull conjecture, which states that for even $n$, every regular $n$-vertex graph with degree at least about $n/2$ has a 1-factorization and was confirmed for large graphs in 2014. Supporting the overfull conjecture as well as generalizing the 1-factorization conjecture in an asymptotic way, in this paper, we show that for any given $0<\varepsilon <1$, there exists a positive integer $n_0$ such that the following statement holds: if $G$ is a graph on $2n\ge n_0$ vertices with minimum degree at least $(1+\varepsilon)n$, then $G$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph.