Determining the vertex stabilizers of 4-valent half-arc-transitive graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917927470497792 |
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| author | Xia, Binzhou Zhang, Zhishuo Zhou, Sanming |
| author_facet | Xia, Binzhou Zhang, Zhishuo Zhou, Sanming |
| contents | We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Marušič and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determining the vertex stabilizers of 4-valent half-arc-transitive graphs Xia, Binzhou Zhang, Zhishuo Zhou, Sanming Combinatorics 05C25, 20B25, 20B35 We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Marušič and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia. |
| title | Determining the vertex stabilizers of 4-valent half-arc-transitive graphs |
| topic | Combinatorics 05C25, 20B25, 20B35 |
| url | https://arxiv.org/abs/2502.12763 |