The core conjecture of Hilton and Zhao

A simple graph G with maximum degree Δ is overfull if $\left|E\right(G\left)\right|>\Delta \lfloor \left|V\right(G\left)\right|/2\rfloor$. The core of G, denoted $G_{\Delta }$, is the subgraph of G induced by its vertices of degree Δ. 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 $\Delta \left(G_{\Delta }\right)\le 2$, then ${\chi }^{\prime }\left(G\right)=\Delta +1$ 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 $\Delta \ge 4$.