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

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 bipartite graphs, independently chosen from $\mathcal{G}(K_{n, n}, p)$, and we obtain a tight bound, up to a factor of $n^{o(1)}$, for the threshold probability at which such graphs attain maximal connectivity. This resolves a conjecture of Alon, Defant, and Kravitz up to lower-order terms. 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.Comment: 30 pages, 9 figure