Abstract
We consider the following matching-based routing problem. Initially, each vertex v of a connected graph G is occupied by a pebble which has a unique destination \(\pi (v)\). In each round the pebbles across the edges of a selected matching in G are swapped, and the goal is to route each pebble to its destination vertex in as few rounds as possible. We show that if G is a sufficiently strong d-regular spectral expander then any permutation \(\pi \) can be achieved in \(O(\log n)\) rounds. This is optimal for constant d and resolves a problem of Alon et al. (SIAM J Discret Math 7:516–530, 1994).
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Nenadov, R. Routing Permutations on Spectral Expanders via Matchings. Combinatorica 43, 737–742 (2023). https://doi.org/10.1007/s00493-023-00033-8
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DOI: https://doi.org/10.1007/s00493-023-00033-8