Long cycles in random subgraphs of graphs with large minimum degree

TL;DR

This paper provides a shorter proof that in large minimum degree graphs, the random subgraph almost surely contains long cycles, extending understanding of cycle lengths in random subgraphs.

Contribution

The authors present a more concise proof of a known result about long cycles in random subgraphs of graphs with large minimum degree, using depth-first search techniques.

Findings

Shorter proof of the existence of long cycles in random subgraphs

With high probability, random subgraphs contain cycles of length proportional to minimum degree

Applicable to graphs with minimum degree at least k

Abstract

Let $G$ be any graph of minimum degree at least $k$, and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Recently, Krivelevich, Lee and Sudakov showed that if $pk\to\infty$ then with probability tending to 1 $G_p$ contains a cycle of length at least $(1-o(1))k$. We give a much shorter proof of this result, also based on depth-first search.