Clique packings in random graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918160359227392 |
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| author | Griffiths, Simon Mattos, Letícia |
| author_facet | Griffiths, Simon Mattos, Letícia |
| contents | We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi random graph $G(n,p)$. Recently Acan and Kahn showed that the largest such family contains only $O(n^2/(\log{n})^3)$ cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, $Ω(n^2/(\log{n})^3)$, by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound $O(n^2/(\log{n})^3)$ and discuss the problem of the precise size of the largest such clique packing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clique packings in random graphs Griffiths, Simon Mattos, Letícia Combinatorics Probability We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi random graph $G(n,p)$. Recently Acan and Kahn showed that the largest such family contains only $O(n^2/(\log{n})^3)$ cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, $Ω(n^2/(\log{n})^3)$, by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound $O(n^2/(\log{n})^3)$ and discuss the problem of the precise size of the largest such clique packing. |
| title | Clique packings in random graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2405.00667 |