Bounds on the bisection width for random d -regular graphs

In this paper we provide an explicit way to compute asymptotically almost sure upper bounds on the bisection width of random d$d$-regular graphs, for any value of d$d$. The upper bounds are obtained from the analysis of the performance of a randomized greedy algorithm to find bisections of d$d$-regular graphs. We provide bounds for 5≤d≤12$5\le d\le 12$. We also give empirical values of the size of the bisection found by the algorithm for some small values of d$d$ and compare them with numerical approximations of our theoretical bounds. Our analysis also gives asymptotic lower bounds for the size of the maximum bisection.