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

具有最大度 Δ 的简单图G是满满的，如果$|乙（G）|>\mathrm{\Delta }\lfloor |V（G）|/2\rfloor$。G 的核心，表示为$G_{\mathrm{\Delta }}$，是G由其 Δ 度顶点导出的子图。显然， G的色指数等于$\mathrm{\Delta }+1$如果G过满。相反，希尔顿和赵在 1996 年猜想，如果G是一个简单的连通图，$\mathrm{\Delta }\ge 3$和$\mathrm{\Delta }（G_{\mathrm{\Delta }}）\le 2$， 然后${\chi }^{\prime }（G）=\mathrm{\Delta }+1$意味着G已满或$G={磷}^{⁎}$， 在哪里${磷}^{⁎}$通过删除顶点从 Petersen 图获得。卡里奥拉罗和卡里奥拉罗解决了基本情况$\mathrm{\Delta }=3$2003 年，Cranston 和 Rabern 证明了下一个案例，$\mathrm{\Delta }=4$，2019年。在这篇论文中，我们为所有人证明了这个猜想$\mathrm{\Delta }\ge 4$。