Given graphs X and Y with vertex sets V(X) and V(Y) of the same
cardinality, we define a graph FS(X,Y) whose vertex set consists of
all bijections Ο:V(X)βV(Y), where two bijections Ο and
Οβ² are adjacent if they agree everywhere except for two adjacent
vertices a,bβV(X) such that Ο(a) and Ο(b) are adjacent in
Y. This setup, which has a natural interpretation in terms of friends and
strangers walking on graphs, provides a common generalization of Cayley graphs
of symmetric groups generated by transpositions, the famous 15-puzzle,
generalizations of the 15-puzzle as studied by Wilson, and work of Stanley
related to flag h-vectors. We derive several general results about the graphs
FS(X,Y) before focusing our attention on some specific choices of
X. When X is a path graph, we show that the connected components of
FS(X,Y) correspond to the acyclic orientations of the complement of
Y. When X is a cycle, we obtain a full description of the connected
components of FS(X,Y) in terms of toric acyclic orientations of the
complement of Y. We then derive various necessary and/or sufficient
conditions on the graphs X and Y that guarantee the connectedness of
FS(X,Y). Finally, we raise several promising further questions.Comment: 28 pages, 5 figure