Year: 2024
Type: preprint
Abstract: 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 $2048λ/η^20$ 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.
Source: arXiv (Cornell University)
Authors: Wang, Xia, Yang, Donglei, Yang, Fan, Yang, Haotian
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404573732