On the Connectivity of Friends-and-strangers Graphs

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 Erdős--Rényi 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 $p_1p_2 \leq (p_0')^2$.

Figures (6)

• Figure 1: The graph $\mathsf{FS}(P_3,P_3)$. The labels on the vertices mark the bijection from the positions to the people the vertex corresponds to, e.g., $312$ represents person $3$ is on position $1,$ person $1$ is on position $2,$ and person $2$ is on position $3.$
• Figure 3: The arrow depiction of an example graph $X.$ Black edges and vertices are part of $X.$ The large blue vertex is $n,$ and the dashed red loops are the blocks of $X.$ The arrow depiction is represented by the blue arrows.
• Figure 4: The vertices $w', u', v',$ and $x'(w)$ and the sets of vertices $P(w)$ and $R(w)$ as well as the corresponding vertices and sets in $Y.$ Black edges represent edges in the graph and red edges represent edges which do not exist in the graph.
• Figure 5: The vertices $w', u', v', h'(w)$ and $x'(w)$ and the sets of vertices $P(w)$ and $R(w)$ as well as the corresponding vertices and sets in $Y.$ Black edges represent edges in the graph and red edges represent edges which do not exist in the graph.
• Figure 6: The vertices $w', u', v', h'(w), q_1'(w), q_2'(w)$ and $x'(w)$ and the sets of vertices $P(w)$ and $R(w)$ as well as the corresponding vertices and sets in $Y.$ Black edges represent edges in the graph and red edges represent edges which do not exist in the graph.

Theorems & Definitions (69)

• Definition 1.1
• Example 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.10
• Proposition 2.1: defant2021friends