On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908545818034176 |
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| author | Weiß, Meike Akpanya, Reymond Niemeyer, Alice C. |
| author_facet | Weiß, Meike Akpanya, Reymond Niemeyer, Alice C. |
| contents | Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the projective plane, the torus and the Klein bottle each are in one-to-one correspondence with certain subgraphs of the dual graph $G^\ast$. Here, we exploit this characterisation and show that two strong embeddings of $G$ on the projective plane, the torus or the Klein bottle are isomorphic if and only if the corresponding subgraphs of $G^{\ast}$ are contained in the same orbit under $\mathrm{Aut}(G^{\ast})$. This allows us to construct a data base containing all isomorphism classes of strong embeddings on the projective plane, the torus and the Klein bottle of all 3-connected cubic planar graphs with up to 22 vertices. Moreover, we establish that cyclically 4-edge connected cubic planar graphs can be strongly embedded on orientable surfaces of positive genera. We use this to show that a 3-connected cubic planar graph has no strong embedding on orientable surfaces of positive genera if and only if it is the dual of an Apollonian network. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces Weiß, Meike Akpanya, Reymond Niemeyer, Alice C. Combinatorics Discrete Mathematics Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the projective plane, the torus and the Klein bottle each are in one-to-one correspondence with certain subgraphs of the dual graph $G^\ast$. Here, we exploit this characterisation and show that two strong embeddings of $G$ on the projective plane, the torus or the Klein bottle are isomorphic if and only if the corresponding subgraphs of $G^{\ast}$ are contained in the same orbit under $\mathrm{Aut}(G^{\ast})$. This allows us to construct a data base containing all isomorphism classes of strong embeddings on the projective plane, the torus and the Klein bottle of all 3-connected cubic planar graphs with up to 22 vertices. Moreover, we establish that cyclically 4-edge connected cubic planar graphs can be strongly embedded on orientable surfaces of positive genera. We use this to show that a 3-connected cubic planar graph has no strong embedding on orientable surfaces of positive genera if and only if it is the dual of an Apollonian network. |
| title | On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2509.14964 |