Long cycles in percolated expanders

TL;DR

This paper proves that in certain percolated expander graphs, long cycles and paths are highly likely to exist, with probabilities approaching certainty as the graph size grows, under specific expansion conditions.

Contribution

It establishes probabilistic guarantees for the existence of long cycles and paths in percolated expanders with given expansion properties, extending understanding of their structural resilience.

Findings

Long cycles of length proportional to the number of vertices exist with high probability.

Paths of linear length are likely in percolated graphs under specified expansion conditions.

Probabilities of not having such long structures decay exponentially with graph size.

Abstract

Given a graph $G$ and probability $p$, we form the random subgraph $G_p$ by retaining each edge of $G$ independently with probability $p$. Given $d\in\mathbb{N}$ and constants $0<c<1, \varepsilon>0$, we show that if every subset $S\subseteq V(G)$ of size exactly $\frac{c|V(G)|}{d}$ satisfies $|N(S)|\ge d|S|$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a cycle of length $\Omega(\varepsilon^2c^2|V(G)|)$ is exponentially small in $|V(G)|$. As an intermediate step, we also show that given $k,d\in \mathbb{N}$ and a constant $\varepsilon>0$, if every subset $S\subseteq V(G)$ of size exactly $k$ satisfies $|N(S)|\ge kd$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a path of length $\Omega(\varepsilon^2 kd)$ is exponentially small. We further discuss applications of these results to $K_{s,t}$-free graphs of maximal density.