On Expansion of Random Regular Graphs: Improved Lower Bounds for Small Even Degrees

TL;DR

This paper improves lower bounds on the expansion of small even degree random regular graphs using a scoring-based tie-breaking method, providing specific probabilistic bounds for degrees 4, 6, and 8.

Contribution

It introduces a simple scoring-based tie-breaking technique that enhances lower bounds on the expansion of random regular graphs with small even degrees.

Findings

Expansion for degree 4 is at least 0.489 with high probability.

Expansion for degree 6 is at least 1.120 with high probability.

Expansion for degree 8 is at least 1.813 with high probability.

Abstract

We show that a simple scoring-based tie-breaking can help improve lower bounds for the expansion (aka isoperimetric number) of random regular graphs with small even degrees. Specifically, for degrees 4, 6 and 8, we show that, with high probability, the expansions are at least 0.489, 1.120 and 1.813 respectively.