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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.19880 |
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| _version_ | 1866912556267864064 |
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| author | Lekše, Maruša Toledo, Micael |
| author_facet | Lekše, Maruša Toledo, Micael |
| contents | In this paper we classify cubic vertex-transitive graphs of girth $7$, based on their signature. Such a graph is either a truncation of an arc-transitive dihedral scheme on a $7$-regular graph, the skeleton of a rotary map of type $\{7,3\}$, a member of an infinite family of Cayley graphs, or is one of the of the generalised Petersen graphs $\text{Pet}(13,5)$, $\text{Pet}(15,4)$, $\text{Pet}(17,4)$ or the Coxeter graph. We show that for a cubic vertex-transitive graphs $Γ$ of girth $7$, if every edge of $Γ$ is contained in the same number of $7$-cycles, then $Γ$ is also arc-transitive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cubic vertex-transitive graphs of girth seven Lekše, Maruša Toledo, Micael Combinatorics In this paper we classify cubic vertex-transitive graphs of girth $7$, based on their signature. Such a graph is either a truncation of an arc-transitive dihedral scheme on a $7$-regular graph, the skeleton of a rotary map of type $\{7,3\}$, a member of an infinite family of Cayley graphs, or is one of the of the generalised Petersen graphs $\text{Pet}(13,5)$, $\text{Pet}(15,4)$, $\text{Pet}(17,4)$ or the Coxeter graph. We show that for a cubic vertex-transitive graphs $Γ$ of girth $7$, if every edge of $Γ$ is contained in the same number of $7$-cycles, then $Γ$ is also arc-transitive. |
| title | Cubic vertex-transitive graphs of girth seven |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.19880 |