Dense minors and bipartite independence numbers
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908652614451200 |
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| author | Wang, Xia Yang, Donglei |
| author_facet | Wang, Xia Yang, Donglei |
| contents | A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{N}$ with $m \le n^{1-\varepsilon}$, every $n$-vertex $m$-joined graph $G$ contains a minor with density $Ω\!\left(\tfrac{n}{\sqrt{m}}\right)$, which is best possible up to a constant factor. When $m \ge n^{1-\varepsilon}$, we further show that $G$ contains a clique minor of order $Ω\!\left(\tfrac{n}{\sqrt{m\log m}}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dense minors and bipartite independence numbers Wang, Xia Yang, Donglei Combinatorics A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{N}$ with $m \le n^{1-\varepsilon}$, every $n$-vertex $m$-joined graph $G$ contains a minor with density $Ω\!\left(\tfrac{n}{\sqrt{m}}\right)$, which is best possible up to a constant factor. When $m \ge n^{1-\varepsilon}$, we further show that $G$ contains a clique minor of order $Ω\!\left(\tfrac{n}{\sqrt{m\log m}}\right)$. |
| title | Dense minors and bipartite independence numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.06656 |