On the Asymmetric Generalizations of Two Extremal Questions on Friends-and-Strangers Graphs

Abstract

For two graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality $n,$ the friends-and-strangers graph $\mathsf{FS}(X,Y)$ was recently defined by Defant and Kravitz. The vertices of $\mathsf{FS}(X,Y)$ are the bijections from $V(X)$ to $V(Y),$ and two bijections $σ$ and $τ$ are adjacent if they agree everywhere except at two vertices $a,b\in V(X)$ such that $a$ and $b$ are adjacent in $X$ and $σ(a)$ and $σ(b)$ are adjacent in $Y.$ We study generalized versions of two problems by Alon, Defant, and Kravitz. First, we show that if $X$ and $Y$ have minimum degrees $δ(X)$ and $δ(Y)$ that satisfy $δ(X)> n/2, δ(Y)>n/2,$ and $2\min(δ(X), δ(Y))+3\max(δ(X), δ(Y))\ge 3n,$ then $\mathsf{FS}(X,Y)$ is connected. As a corollary, we settle a recent conjecture by Alon, Defant, and Kravitz stating that there exists a number $d_n = 3n/5 + O(1)$ such that if both $X$ and $Y$ have minimum degrees at least $d_n,$ the graph $\mathsf{FS}(X,Y)$ is connected. When $X$ and $Y$ are bipartite, a parity obstruction prevents $\mathsf{FS}(X,Y)$ from being connected. We show that if $X$ and $Y$ are edge-subgraphs of $K_{r,r}$ that satisfy $δ(X)+δ(Y)\ge 3r/2+1,$ then the graph $\mathsf{FS}(X,Y)$ has exactly two connected components. As a corollary, we provide an almost complete answer to another recent question of Alon, Defant, and Kravitz asking for the minimum number $d^*_{r,r}$ such that for any edge-subgraph $X$ of $K_{r,r}$ satisfying $δ(X)\ge d^*_{r,r},$ the graph $\mathsf{FS}(X,K_{r,r})$ has exactly two connected components. We show that $d^*_{r,r} = r/2+1$ when $r$ is even and $d^*_{r,r}\in \{\lceil r/2\rceil, \lceil r/2\rceil+1\}$ when $r$ is odd.