A large hole in pseudo-random graphs

TL;DR

This paper proves the existence of large induced cycles in certain pseudo-random graphs and shows that the number of non-isomorphic induced subgraphs is exponentially large, establishing bounds that are tight up to constants.

Contribution

It establishes bounds on the size of large induced cycles in $(n,d,rac{ ext{spectral gap}}{d})$-graphs and quantifies the diversity of induced subgraphs, with results shown to be optimal up to constants.

Findings

Existence of large induced cycles proportional to $n/d$

Exponential lower bound on the number of non-isomorphic induced subgraphs

Bounds are tight up to a constant factor

Abstract

We show that there exist constants $\delta_1,\delta_2>0$ such that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le\delta_1$, then $G$ contains an induced cycle of length at least $\delta_2n/d$. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such $G$ is at least exponential in $n\log d/d$, and further demonstrate that this is tight up to a constant factor in the exponent.