Proof of the Core Conjecture of Hilton and Zhao

Let $G$ be a simple graph with maximum degree $Δ$. We call $G$ \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. A classic result of Vizing shows that $χ'(G)$, the chromatic index of $G$, is either $Δ$ or $Δ+1$. It is NP-complete to determine the chromatic index for a general graph. However, if $G$ is overfull then $χ'(G)=Δ+1$. Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ if and only if $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. This conjecture, if true, implies an easy approach for calculating $χ'(G)$ for graphs $G$ satisfying the conditions. The progress on the conjecture has been slow: it was only confirmed for $Δ=3,4$, respectively, in 2003 and 2017. In this paper, we confirm this conjecture for all $Δ\ge 4$.

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