Connectivity of Old and New Models of Friends-and-Strangers Graphs

TL;DR

This paper studies the connectivity thresholds of friends-and-strangers graphs derived from random graphs, improving bounds, disproving conjectures, and generalizing the concept to graphs with vertex multiplicities.

Contribution

It refines the bounds on connectivity thresholds for friends-and-strangers graphs, disproves existing conjectures, and introduces a generalized framework with vertex multiplicities.

Findings

Improved lower bounds on connectivity thresholds.

Disproved two conjectures by Alon, Defant, and Kravitz.

Established a tight upper bound for random bipartite graphs.

Abstract

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the threshold probability at which such graphs attain maximal connectivity. We slightly improve the lower bounds on the threshold probabilities, thus disproving two conjectures of Alon, Defant and Kravitz. We also improve the upper bound on the threshold probability in the case of random bipartite graphs, and obtain a tight bound up to a factor of $n^{o(1)}$. Further, we introduce a generalization of the notion of friends-and-strangers graphs in which vertices of the starting graphs are allowed to have multiplicities and obtain generalizations of previous results of Wilson and of Defant and Kravitz in this new setting.