Expander spanning subgraphs with large girth

TL;DR

This paper proves a conjecture that certain highly connected regular graphs contain spanning subgraphs with large girth proportional to their diameter, using probabilistic methods.

Contribution

It establishes the conjecture for regular expanders with large expansion, advancing understanding of graph structure and girth properties.

Findings

Proved the conjecture for regular expander graphs with large expansion

Used the Local Lemma in the proof

Demonstrated existence of spanning subgraphs with large girth

Abstract

We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.