Cayley graphs on elementary abelian groups of extreme degree have complete cores
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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_ | 1866912210948718592 |
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| author | Rao, Guang Tan, Colin |
| author_facet | Rao, Guang Tan, Colin |
| contents | Nešetřil and Šámal asked whether every cubelike graph has a cubelike core. Mančinska, Pivotto, Roberson and Royle answered this question in the affirmative for cubelike graphs whose core has at most $32$ vertices. When the core of a cubelike graph has at most $16$ vertices, they gave a list of these cores, from which it follows that every cubelike graph with degree strictly less than $5$ has a complete core. We prove the following extension: if the degree of a cubelike graph is either strictly less than $5$ or at least $5$ less than the number of its vertices, then its core is complete and induced by a $\mathbb{F}_2$-vector subspace of its vertices. Thus we also answer Nešetřil and Šámal's question in the affirmative for cubelike graphs with degree at least $5$ less than the number of vertices. Our result is sharp as the $5$-regular folded $5$-cube and its graph complement are both non-complete cubelike graph cores. We also prove analogous results for Cayley graphs on elementary abelian $p$-groups for odd primes $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cayley graphs on elementary abelian groups of extreme degree have complete cores Rao, Guang Tan, Colin Combinatorics Primary 05C50, 05C60, Secondary 20K25 Nešetřil and Šámal asked whether every cubelike graph has a cubelike core. Mančinska, Pivotto, Roberson and Royle answered this question in the affirmative for cubelike graphs whose core has at most $32$ vertices. When the core of a cubelike graph has at most $16$ vertices, they gave a list of these cores, from which it follows that every cubelike graph with degree strictly less than $5$ has a complete core. We prove the following extension: if the degree of a cubelike graph is either strictly less than $5$ or at least $5$ less than the number of its vertices, then its core is complete and induced by a $\mathbb{F}_2$-vector subspace of its vertices. Thus we also answer Nešetřil and Šámal's question in the affirmative for cubelike graphs with degree at least $5$ less than the number of vertices. Our result is sharp as the $5$-regular folded $5$-cube and its graph complement are both non-complete cubelike graph cores. We also prove analogous results for Cayley graphs on elementary abelian $p$-groups for odd primes $p$. |
| title | Cayley graphs on elementary abelian groups of extreme degree have complete cores |
| topic | Combinatorics Primary 05C50, 05C60, Secondary 20K25 |
| url | https://arxiv.org/abs/2501.18297 |