Edge coloring graphs with large 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 $0<\epsilon <1$, 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 $(1+\epsilon )n$, then $G$ has chromatic index $\Delta (G)$ if and only if $G$ contains no overfull subgraph.

Fixed some typos and revised Lemmas 2.7, 2.11, and 2.12. arXiv admin note: substantial text overlap with arXiv:2104.06253