Long paths and cycles in random subgraphs of H-free graphs

TL;DR

This paper extends classical results on random subgraphs of dense graphs, showing that for certain H-free graphs with high minimum degree, the random subgraph likely contains long cycles proportional to the minimum degree, under specific probability conditions.

Contribution

It generalizes known theorems by establishing the presence of long cycles in random subgraphs of H-free graphs with high minimum degree, depending on the edge retention probability.

Findings

Random subgraphs contain long cycles with high probability when p ≥ (1+ε)/k.

Cycle length is at least linear in the minimum degree k.

Results extend classical theorems to H-free graphs with forbidden subgraphs.

Abstract

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Koml\'os, and Szemer\'edi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \ge \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_{\mathcal{H}}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.