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

Abstract

In this paper we give new asymptotically almost sure lower and upper bounds on the bisection width of random $3-$regular graphs. The main contribution is a new lower bound on the bisection width of $0.103295n$, based on a first moment method together with a structural decomposition of the graph, thereby improving a 27 year old result of Kostochka and Melnikov. We also give a complementary upper bound of $0.139822n$, combining known spectral ideas with original combinatorial insights. Developping further this approach, with the help of Monte Carlo simulations, we obtain a non-rigorous upper bound of $0.131366n$.Comment: 48 pages, 20 figure