On the Connectivity of Friends-and-strangers Graphs

TL;DR

This paper investigates the connectivity properties of friends-and-strangers graphs, establishing conditions for their k-connectivity in various graph configurations, including complete, star, and Erdős–Rényi random graphs, with tight bounds and probabilistic thresholds.

Contribution

It provides new bounds and probabilistic thresholds for the k-connectivity of friends-and-strangers graphs across different graph models, extending previous connectivity results.

Findings

Connectivity equals minimum degree for complete and star graphs.

Asymptotic conditions for k-connectivity in Erdős–Rényi random graphs.

Thresholds for disconnection in random graph models.

Abstract

Friends-and-strangers graphs, coined by Defant and Kravitz, are denoted by $\mathsf{FS}(X,Y)$ where $X$ and $Y$ are both graphs on $n$ vertices. The graph $X$ represents positions and edges mark adjacent positions while the graph $Y$ represents people and edges mark friendships. The vertex set of $\mathsf{FS}(X,Y)$ consists of all one-to-one placements of people on positions, and there is an edge between any two placements if it is possible to swap two people who are friends and on adjacent positions to get from one placement to the other. Previous papers have studied when $\mathsf{FS}(X,Y)$ is connected. In this paper, we consider when $\mathsf{FS}(X,Y)$ is $k$-connected where a graph is $k$-connected if it remains connected after removing any $k-1$ or less vertices. We first consider $\mathsf{FS}(X,Y)$ when $Y$ is a complete graph or star graph. We find tight bounds on their connectivity, proving their connectivity equals their minimum degree. We further consider the size of the connected components of $\mathsf{FS}(X,\mathsf{Star}_n)$ where $X$ is connected. We show that asymptotically similar conditions as the conditions mentioned by Bangachev are sufficient for $\mathsf{FS}(X,Y)$ to be $k$-connected. Finally, we consider when $X$ and $Y$ are independent Erd\H{o}s--R\'enyi random graphs on $n$ vertices and edge probability $p_1$ and $p_2,$ respectively. We show that for $p_0 = n^{-1/2+o(1)},$ if $p_1p_2\geq p_0^2$ and $p_1,$ $p_2 \geq w(n) p_0$ where $w(n) \rightarrow 0$ as $n \rightarrow \infty,$ then $\mathsf{FS}(X,Y)$ is $k$-connected with high probability. This is asymptotically tight as we show that below an asymptotically similar threshold $p_0'=n^{-1/2+o(1)}$, the graph $\mathsf{FS}(X,Y)$ is disconnected with high probability if $p_1p_2 \leq (p_0')^2$.