A conjecture on the maximum cut and bisection width in random regular graphs

Abstract

The asymptotic properties of random regular graphs are objects of extensive study inmathematics and physics. In this paper we argue, using the theory of spin glasses inphysics, that in random regular graphs the maximum cut size asymptotically equals thenumber of edges in the graph minus the minimum bisection size. Maximum cut andminimal bisection are two famous NP-complete problems with no known general relationbetween them; hence our conjecture is a surprising property for random regular graphs. Wefurther support the conjecture with numerical simulations. A rigorous proof of this relationis an obvious challenge.