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arXiv · 2108.04399

The Core Conjecture of Hilton and Zhao II: a Proof

Yan Cao·Guantao Chen·Guangming Jing·Songling Shan
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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 $\Delta\ge 4$.

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Yan Cao, Guantao Chen, Guangming Jing, Songling Shan. 2021-08-10. The Core Conjecture of Hilton and Zhao II: a Proof. https://arxiv.org/abs/2108.04399

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