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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.07028 |
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| _version_ | 1866913539408527360 |
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| author | Pan, Xiang-Feng Mao, Jing-Zhong Liu, Hui-Qing |
| author_facet | Pan, Xiang-Feng Mao, Jing-Zhong Liu, Hui-Qing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07028 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The proof of a conjecture about cages Pan, Xiang-Feng Mao, Jing-Zhong Liu, Hui-Qing Combinatorics 05C40 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. |
| title | The proof of a conjecture about cages |
| topic | Combinatorics 05C40 |
| url | https://arxiv.org/abs/2410.07028 |