The Core Conjecture of Hilton and Zhao II: a Proof

TL;DR

This paper proves Hilton and Zhao's conjecture that for all graphs with maximum degree at least 4, if the chromatic index equals the maximum degree plus one and the core has maximum degree at most 2, then the graph is either overfull or a Petersen-derived graph.

Contribution

The paper confirms Hilton and Zhao's conjecture for all graphs with maximum degree at least 4, extending previous results for degrees 3 and 4.

Findings

Confirmed the conjecture for all $oldsymbol{ ext{Δ} oldsymbol{ extgeq} 4}$.

Established the characterization of graphs with $oldsymbol{ ext{χ'}(G) = ext{Δ} + 1}$ under core degree constraints.

Extended the understanding of overfull graphs and their relation to the chromatic index.

Abstract

A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case $\Delta=3$ in 2003, and Cranston and Rabern proved the next case $\Delta=4$ in 2019. In this paper, we give a proof of this conjecture for all $\Delta\ge 4$.