Routing Permutations on Spectral Expanders via Matchings

We consider the following matching-based routing problem. Initially, each vertex v of a connected graph G is occupied by pebble which has unique destination $$\pi (v)$$ . In round pebbles across edges selected matching in are swapped, and goal to route its as few rounds possible. show that if sufficiently strong d-regular spectral expander then any permutation $$ can be achieved $$O(\log n)$$ rounds. This optimal for constant d resolves problem Alon et al. (SIAM J Discret Math 7:516–530, 1994).