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arXiv · 2410.07028

The proof of a conjecture about cages

Xiang-Feng Pan·Jing-Zhong Mao·Hui-Qing Liu
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Abstract

The girth of a graph is defined as the length of a shortest cycle in the graph. A $(k; g)$-cage is a graph of minimum order among all $k$-regular graphs with girth $g$. A cycle $C$ in a graph $G$ is termed nonseparating if the graph $G-V(C)$ remains connected. A conjecture, proposed in [T. Jiang, D. Mubayi. Connectivity and Separating Sets of Cages. J. Graph Theory 29(1)(1998) 35--44], posits that every cycle of length $g$ within a $(k; g)$-cage is nonseparating. While the conjecture has been proven for even $g$ in the aforementioned work, this paper presents a proof demonstrating that the conjecture holds true for odd $g$ as well. Thus, the previously mentioned conjecture was proven to be true.

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Xiang-Feng Pan, Jing-Zhong Mao, Hui-Qing Liu. 2024-10-09. The proof of a conjecture about cages. https://arxiv.org/abs/2410.07028

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