The maximum number of $k$-cliques of 7-connected 1-planar graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913098945789952 |
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| author | Huang, Yuanqiu Zhang, Licheng |
| author_facet | Huang, Yuanqiu Zhang, Licheng |
| contents | In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the high-connectivity end, we prove that every $n$-vertex 7-connected 1-planar graph has at most $4n-12$ edges, $4n-16$ triangles, and $n-6$ copies of $K_4$. Hence the total number of cliques is at most $10n-33$. All bounds are sharp for infinitely many values of $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06378 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The maximum number of $k$-cliques of 7-connected 1-planar graphs Huang, Yuanqiu Zhang, Licheng Combinatorics In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the high-connectivity end, we prove that every $n$-vertex 7-connected 1-planar graph has at most $4n-12$ edges, $4n-16$ triangles, and $n-6$ copies of $K_4$. Hence the total number of cliques is at most $10n-33$. All bounds are sharp for infinitely many values of $n$. |
| title | The maximum number of $k$-cliques of 7-connected 1-planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.06378 |