Topological cliques in sparse expanders
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929597707190272 |
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| author | Wang, Xia Yang, Donglei Yang, Fan Yang, Haotian |
| author_facet | Wang, Xia Yang, Donglei Yang, Fan Yang, Haotian |
| contents | In the paper, we focus on embedding clique immersions and subdivisions within sparse expanders, and we derive the following main results: (1) For any $0< η< 1/2$, there exists $K>0$ such that for sufficiently large $n$, every $(n,d,λ)$-graph $G$ contains a $K_{(1-5η)d}$-immersion when $d\geq Kλ$. (2) For any $\varepsilon>0$ and $0<η<1/2$, the following holds for sufficiently large $n$. Every $(n,d,λ)$-graph $G$ with $2048λ/η^2<d\leq ηn^{1/2-\varepsilon}$ contains a $K_{(1-η)d}^{(\ell)}$-subdivision, where $\ell = 2 \left\lceil \log(η^2n/4096)\right\rceil + 5$. (3) There exists $c>0$ such that the following holds for sufficiently large $d$. If $G$ is an $n$-vertex graph with average degree $d(G)\geq d$, then $G$ contains a $K_{c d}^{(\ell)}$-immersion for some $\ell\in \mathbb{N}$.
In 2018, Dvo{ř}{á}k and Yepremyan asked whether every graph $G$ with $δ(G)\geq t$ contains a $K_t$-immersion. Our first result shows that it is asymptotically true for $(n,d,λ)$-graphs when $λ=o(d)$. In addition, our second result extends a result of Dragani{ć}, Krivelevich and Nenadov on balanced subdivisions. The last result generalises a result of DeVos, Dvo{ř}{á}k, Fox, McDonald, Mohar, Scheide on $1$-immersions of large cliques in dense graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological cliques in sparse expanders Wang, Xia Yang, Donglei Yang, Fan Yang, Haotian Combinatorics In the paper, we focus on embedding clique immersions and subdivisions within sparse expanders, and we derive the following main results: (1) For any $0< η< 1/2$, there exists $K>0$ such that for sufficiently large $n$, every $(n,d,λ)$-graph $G$ contains a $K_{(1-5η)d}$-immersion when $d\geq Kλ$. (2) For any $\varepsilon>0$ and $0<η<1/2$, the following holds for sufficiently large $n$. Every $(n,d,λ)$-graph $G$ with $2048λ/η^2<d\leq ηn^{1/2-\varepsilon}$ contains a $K_{(1-η)d}^{(\ell)}$-subdivision, where $\ell = 2 \left\lceil \log(η^2n/4096)\right\rceil + 5$. (3) There exists $c>0$ such that the following holds for sufficiently large $d$. If $G$ is an $n$-vertex graph with average degree $d(G)\geq d$, then $G$ contains a $K_{c d}^{(\ell)}$-immersion for some $\ell\in \mathbb{N}$. In 2018, Dvo{ř}{á}k and Yepremyan asked whether every graph $G$ with $δ(G)\geq t$ contains a $K_t$-immersion. Our first result shows that it is asymptotically true for $(n,d,λ)$-graphs when $λ=o(d)$. In addition, our second result extends a result of Dragani{ć}, Krivelevich and Nenadov on balanced subdivisions. The last result generalises a result of DeVos, Dvo{ř}{á}k, Fox, McDonald, Mohar, Scheide on $1$-immersions of large cliques in dense graphs. |
| title | Topological cliques in sparse expanders |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.12237 |