Routing permutations on graphs via matchings

We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a “pebble” which has a unique destination π(v) in G (so that π is a permutation of the vertices of G). It is required to route all the pebbles to their respective destinations by performing a sequence of moves of the following type: A disjoint set of edges is selected and the pebbles at each edge’s endpoints are interchanged. The problem of interest is to minimize the number of steps required for any possible permutation π. In this paper we investigate this routing problem for a variety of graphs G, including trees, complete graphs, hypercubes, Cartesian products of graphs, expander graphs and Cayley graphs. In addition, we relate this routing problem to certain network flow problems, and to several graph invariants including diameter, eigenvalues and expansion coefficients.