3 research outputs found
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 , and we obtain a
tight bound, up to a factor of , 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