Topological Minors in Typical Lifts

TL;DR

This paper investigates the size of the largest topological clique in random lifts of complete graphs, confirming a conjecture about the existence of large clique subdivisions when the lift size is proportional to the number of vertices.

Contribution

It proves a conjecture that large clique subdivisions appear in random lifts of complete graphs when the lift size is linear in the number of vertices, and establishes tight bounds for various lift sizes.

Findings

For  \, ext{lift size} \, ext{ ext{at least linear in } n}, a clique subdivision of size n exists almost surely.

For  \, ext{lift size} \, ext{ ext{less than linear in } n}, such subdivisions almost surely do not exist.

The size of the largest clique subdivision in smaller lifts is approximately  \, (1 - o(1)) \, ext{ ext{times}} \, \\sqrt{rac{2n \, ext{lift size}}{1 - 1/ ext{lift size}}}.

Abstract

An $\ell$-lift of a graph $G$ is any graph obtained by replacing every vertex of $G$ with an independent set of size $\ell$, and connecting every pair of two such independent sets that correspond to an edge in $G$ by a matching of size $\ell$. Graph lifts have found numerous interesting applications and connections to a variety of areas over the years. Of particular importance is the random graph model obtained by considering an $\ell$-lift of a graph sampled uniformly at random. This model was first introduced by Amit and Linial in 1999, and has been extensively investigated since. In this paper, we study the size of the largest topological clique in random lifts of complete graphs. In 2006, Drier and Linial raised the conjecture that almost all $\ell$-lifts of the complete graph on $n$ vertices contain a subdivision of a clique of order $\Omega(n)$ as a subgraph provided $\ell$ is at least linear in $n$. We confirm their conjecture in a strong form by showing that for $\ell \ge (1+o(1))n$, one can almost surely find a subdivision of a clique of order $n$. We prove that this is tight by showing that for $\ell \le (1-o(1))n$, almost all $\ell$-lifts do not contain subdivisions of cliques of order $n$. Finally, for $2 \le \ell \ll n$, we show that almost all $\ell$-lifts of $K_n$ contain a subdivision of a clique on $(1-o(1))\sqrt{\frac{2n \ell}{1-1/\ell}}$ vertices and that this is tight up to the lower order term.