Routing permutations on graphs via matchings

Abstract

A class of routing problems on connected graphs $G$ is considered. Initially, each vertex $v$ of $G$ is occupied by a ``pebble'' that has a unique destination $\pi (v)$ in $G$ (so that $\pi$ is a permutation of the vertices of $G$). It is required that all the pebbles be routed 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 $\pi$. This paper investigates 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, this routing problem is related to certain network flow problems, and to several graph invariants including diameter, eigenvalues, and expansion coefficients.