Chromatic index of dense quasirandom graphs

TL;DR

This paper proves that dense quasirandom graphs with odd order satisfy a conjecture relating their chromatic index to the absence of overfull subgraphs, extending previous results for even order graphs.

Contribution

It confirms Glock, K"uhn, and Osthus's conjecture for dense quasirandom graphs with odd order, completing the understanding for all dense quasirandom graphs.

Findings

Confirmed the conjecture for odd order dense quasirandom graphs.

Extended previous results from even to odd order graphs.

Provided a complete characterization of chromatic index in this class.

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 $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K\"{u}hn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, K\"{u}hn and Osthus is affirmative.