Topological cliques in sparse expanders

TL;DR

This paper investigates embedding large clique immersions and subdivisions in sparse expander graphs, providing new bounds and extending previous results in graph theory related to clique immersions and subdivisions.

Contribution

It introduces new bounds for clique immersions and subdivisions in sparse expanders, extending prior work on clique immersions and balanced subdivisions.

Findings

Every sufficiently large $(n,d, ext{lambda})$-graph contains a large clique immersion.

Sparse expanders contain large clique subdivisions under certain degree conditions.

Dense graphs with high average degree contain large clique immersions.

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 $0< \eta< 1/2$, there exists $K>0$ such that for sufficiently large $n$, every $(n,d,\lambda)$-graph $G$ contains a $K_{(1-5\eta)d}$-immersion when $d\geq K\lambda$. (2) For any $\varepsilon>0$ and $0<\eta <1/2$, the following holds for sufficiently large $n$. Every $(n,d,\lambda)$-graph $G$ with $2048\lambda/\eta^2<d\leq \eta n^{1/2-\varepsilon}$ contains a $K_{(1-\eta)d}^{(\ell)}$-subdivision, where $\ell = 2 \left\lceil \log(\eta^2n/4096)\right\rceil + 5$. (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{\v{r}}{\'a}k and Yepremyan asked whether every graph $G$ with $\delta(G)\geq t$ contains a $K_t$-immersion. Our first result shows that it is asymptotically true for $(n,d,\lambda)$-graphs when $\lambda=o(d)$. In addition, our second result extends a result of Dragani{\'c}, Krivelevich and Nenadov on balanced subdivisions. The last result generalises a result of DeVos, Dvo{\v{r}}{\'a}k, Fox, McDonald, Mohar, Scheide on $1$-immersions of large cliques in dense graphs.