On the minimum bisection of random $3$-regular graphs

TL;DR

This paper establishes new bounds on the minimum bisection width of random 3-regular graphs, improving previous results through probabilistic and combinatorial methods, with both theoretical and simulation-based insights.

Contribution

It provides the first new lower bound in 27 years and refines upper bounds using a combination of analytical and Monte Carlo simulation techniques.

Findings

Lower bound of 0.103295n on bisection width

Upper bound of 0.139822n on bisection width

Non-rigorous improved upper bound via Monte Carlo simulations

Abstract

In this paper we give new bounds on the bisection width of random 3-regular graphs on $n$ vertices. The main contribution is a new lower bound of $0.103295n$ based on a first moment method together with a structural analysis of the graph, thereby improving a 27-year-old result of Kostochka and Melnikov. We also give a complementary upper bound of $0.139822n$ by combining a result of Lyons with original combinatorial insights. Developping this approach further, we obtain a non-rigorous improved upper bound with the help of Monte Carlo simulations.