Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $δ\ge 6$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916546191818752 |
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| author | Limbach, Anna M. Winter, Martin |
| author_facet | Limbach, Anna M. Winter, Martin |
| contents | The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have non-empty intersection in $G$. We say that $G$ is clique convergent if $k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree $δ\ge 6$, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called "triangular-shaped graphs". |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_00503 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $δ\ge 6$ Limbach, Anna M. Winter, Martin Combinatorics 05C69, 57Q15, 57M10, 37E25 The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have non-empty intersection in $G$. We say that $G$ is clique convergent if $k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree $δ\ge 6$, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called "triangular-shaped graphs". |
| title | Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $δ\ge 6$ |
| topic | Combinatorics 05C69, 57Q15, 57M10, 37E25 |
| url | https://arxiv.org/abs/2305.00503 |