  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 $\sigma$ and $\tau$
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 $\sigma(a)$ and $\sigma(b)$ 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
$\delta(X)$ and $\delta(Y)$ that satisfy $\delta(X)> n/2, \delta(Y)>n/2,$ and
$2\min(\delta(X), \delta(Y))+3\max(\delta(X), \delta(Y))\ge 3n,$ 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
$\delta(X)+\delta(Y)\ge 3r/2+1,$ 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
$\delta(X)\ge d^*_{r,r},$ 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.
