Embedding a Praeger-Xu graph into a surface
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913922936733696 |
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| author | Ding, Zhaochen Guo, Zheng Liu, Luyi |
| author_facet | Ding, Zhaochen Guo, Zheng Liu, Luyi |
| contents | Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\operatorname{C}(p,r,s)$, for any odd prime $p$ that does not divide $r$. Our main result establishes a one-to-one correspondence between the isomorphism classes of these maps and the multiplicity-free representations of the dihedral group $\operatorname{D}_{2r}$ over the finite field $\mathbb{F}_p$. This work extends a recent classification for the case where $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Embedding a Praeger-Xu graph into a surface Ding, Zhaochen Guo, Zheng Liu, Luyi Combinatorics Group Theory 05C25, 20B05, 20C15 Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\operatorname{C}(p,r,s)$, for any odd prime $p$ that does not divide $r$. Our main result establishes a one-to-one correspondence between the isomorphism classes of these maps and the multiplicity-free representations of the dihedral group $\operatorname{D}_{2r}$ over the finite field $\mathbb{F}_p$. This work extends a recent classification for the case where $p=2$. |
| title | Embedding a Praeger-Xu graph into a surface |
| topic | Combinatorics Group Theory 05C25, 20B05, 20C15 |
| url | https://arxiv.org/abs/2507.01716 |