Abstract
Given graphs \(X\) and \(Y\) with vertex sets \(V(X)\) and \(V(Y)\) of the same cardinality, we define a graph \(\mathsf{FS}(X,Y)\) whose vertex set consists of all bijections \(\sigma\colon V(X)\to V(Y)\), where two bijections \(\sigma\) and \(\sigma'\) are adjacent if they agree everywhere except for two adjacent vertices \(a,b \in V(X)\) such that \(\sigma(a)\) and \(\sigma(b)\) 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 \(\mathsf{FS}(X,Y)\) before focusing our attention on some specific choices of \(X\). When \(X\) is a path graph, we show that the connected components of \(\mathsf{FS}(X,Y)\) 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 \(\mathsf{FS}(X,Y)\) 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 \(\mathsf{FS}(X,Y)\). Finally, we raise several promising further questions.Mathematics Subject Classifications: 05C40, 05C38, 05A05
Highlights
Let X be a simple graph with n vertices
We prove that FS(X, Y ) is disconnected whenever X and Y both have cut vertices, and we provide a lower bound for the number of connected components
We completely describe the connected components of FS(Cyclen, Y ), where Cyclen is the cycle graph with vertex set [n] and edge set {{i, i + 1} : 1 i n − 1} ∪ {{n, 1}}. This description is much more involved than the description of the connected components of FS(Pathn, Y ); it makes use of toric acyclic orientations, which have appeared in many contexts and were formalized in [10]
Summary
Let X be a simple graph with n vertices. Imagine that n different people, any two of whom are either friends or strangers, are standing so that one person is at each vertex of X. This description is much more involved than (yet very similar in flavor to) the description of the connected components of FS(Pathn, Y ); it makes use of toric acyclic orientations, which have appeared in many contexts and were formalized in [10].
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