Maximum cut and maximum bisection in random regular graphs

We prove that for every fixed degree d, maximum cut and maximum bisection have the same limiting expected density in a uniformly random simple d-regular graph. The proof uses Huang's prescribed-degree interpolation method. Its key input is a structural lemma showing that, for any fixed family of cuts, the single-edge increments of the maximum form the separation matrix of a partition. We apply the interpolation to compare a configuration-model graph on 2n vertices with the disjoint union of two independent n-vertex configuration-model graphs; concentration and conditioning on simplicity then complete the argument.