On the asymmetric generalizations of two extremal questions on friends-and-strangers graphs

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