Conjecture on the maximum cut and bisection width in random regular graphs

TL;DR

This paper conjectures a surprising asymptotic relation between maximum cut size and minimum bisection size in random regular graphs, supported by spin glass theory and numerical simulations, highlighting a potential new insight into these NP-complete problems.

Contribution

It proposes a novel conjecture linking maximum cut and bisection width in random regular graphs, supported by theoretical insights and numerical evidence.

Findings

Conjecture that max cut size asymptotically equals total edges minus min bisection size.

Numerical simulations support the conjecture.

Highlights a potential fundamental relation between two NP-complete problems.

Abstract

Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no known general relation between them, hence our conjecture is a surprising property of random regular graphs. We further support the conjecture with numerical simulations. A rigorous proof of this relation is obviously a challenge.