Year: 2026
Type: preprint
Abstract: We prove that, for every fixed degree $d$ and a uniformly random simple $d$-regular graph $G_{n,d}$ on an even number $n>d$ of vertices, $$\mathbb{E}\operatorname{MaxCut}(G_{n,d})-\mathbb{E}\operatorname{MaxBis}(G_{n,d})=o(n).$$ In other words, requiring the two sides of a cut to have exactly the same size changes the expected optimal cut by only a sublinear number of edges. We prove this first for the configuration model by comparing cuts of each possible cardinality on an $n$-vertex graph with bisections of a related graph on $2n$ vertices. An important technical tool is Huang's interpolation theorem (Huang, 2018); to apply it, we establish a structural property of the change in the optimal cut when a single edge is added. This comparison, together with concentration estimates and Huang's convergence theorem for maximum bisection, shows that maximum cut and maximum bisection have the same limiting density. Conditioning the configuration model on being simple then gives the result for uniformly random simple regular graphs.
Source: arXiv (Cornell University)
DOI: 10.17615/hney-ea67
Authors: Patrick Lopatto, P. M. Aronow
Institution: University of North Carolina at Chapel Hill
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172029669