The proof of a conjecture about cages
Abstract
The girth of a graph is defined as the length of a shortest cycle in the graph. A -cage is a graph of minimum order among all -regular graphs with girth . A cycle in a graph is termed nonseparating if the graph 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 within a -cage is nonseparating. While the conjecture has been proven for even in the aforementioned work, this paper presents a proof demonstrating that the conjecture holds true for odd as well. Thus, the previously mentioned conjecture was proven to be true.
Keywords
Cite
@article{arxiv.2410.07028,
title = {The proof of a conjecture about cages},
author = {Xiang-Feng Pan and Jing-Zhong Mao and Hui-Qing Liu},
journal= {arXiv preprint arXiv:2410.07028},
year = {2024}
}
Comments
This paper was published in Chinese on April 25, 2001, in Volume 14, Issue 2 of the "MATHEMATICA APPLICATA" on pages 99 to 102. This is the English version of the paper, incorporating additional details and rectifying certain typographical errors