Friendly bisections of random graphs

Abstract

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph G(n, 1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n − o(n) vertices have at least as many neighbours in their own part as across. The engine of our proof is a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.

(Dis)assortative partitions on random regular graphs

“…We give empirical evidence that indeed assortative partitions at H = d/2 are found easily and so are disassortative ones. This result has been established at d = n/2 in [Fer+21]. Future work should aim at a rigorous analysis of the related algorithms (or other ones) establishing that indeed these partitions can be found efficiently on random d-regular graphs for all d, or even for all general (perhaps large girth) d-regular graphs.…”

“…For example, there exists an algorithm for d 2 -assortative partitions on dense ER graphs that provably finds them with high probability [Fer+21]. This algorithm uses a greedy node swapping operation to iteratively approach a solution.…”

“…This broad variety of origins created an abundance of terminology in the field. Therefore, results on assortative partitions can be found under the name of cohesive subsets [Mor00], satisfactory partitions [BTV10], friendly partitions [Fer+21], generalized matching cuts [GS21] and local minimum bisection [AMS21]. Likewise, disassortative partitions relate to co-satisfactory partitions [BTV10], unfriendly partitions [Fer+21] and local maximum cut [AMS21].…”

“…Therefore, results on assortative partitions can be found under the name of cohesive subsets [Mor00], satisfactory partitions [BTV10], friendly partitions [Fer+21], generalized matching cuts [GS21] and local minimum bisection [AMS21]. Likewise, disassortative partitions relate to co-satisfactory partitions [BTV10], unfriendly partitions [Fer+21] and local maximum cut [AMS21]. We chose the term assortative, since it describes the phenomenon of similar nodes preferentially attaching to one another.…”

A note on internal partitions: the $5$-regular case and beyond

“…Linial and Louis proved [18] that for every positive integer r, asymptotically almost every 2r-regular graph has an internal partition. Very recently, Ferber et al resolved [11] a conjecture of Füredi by proving that with high probability, the random graph G(n, 1/2) admits a partition of its vertex set into two parts whose sizes differ by at most one in which n−o(n) vertices have at least as many neighbours in their own part as across.…”