Skip to main content
Log in

Routing Permutations on Spectral Expanders via Matchings

Combinatorica Aims and scope Submit manuscript

Cite this article


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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. Aggarwal, A., Bar-Noy, A., Coppersmith, D., Ramaswami, R., Schieber, B., Sudan, M.: Efficient routing in optical networks. J. ACM 43(6), 973–1001 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Chung, F.R.K., Graham, R.L.: Routing permutations on graphs via matchings. SIAM J. Discret. Math. 7(3), 513–530 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Spencer, J.H.: The probabilistic method. In: Wiley-Interscience Series in Discrete Mathematics and Optimization, 4th edn. Wiley, Hoboken (2016)

  4. Broder, A.Z., Frieze, A.M., Upfal, E.: Existence and construction of edge-disjoint paths on expander graphs. SIAM J. Comput. 23(5), 976–989 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Draganić, N., Krivelevich, M., Nenadov, R.: Rolling backwards can move you forward: on embedding problems in sparse expanders. Trans. Am. Math. Soc. 375(7), 5195–5216 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Feldman, P., Friedman, J., Pippenger, N.: Wide-sense nonblocking networks. SIAM J. Discret. Math. 1(2), 158–173 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Friedman, J., Pippenger, N.: Expanding graphs contain all small trees. Combinatorica 7, 71–76 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Glebov, R.: On Hamilton cycles and other spanning structures. PhD thesis, Freie Universität Berlin (2013)

  9. Hoory, S., Linial, N., Widgerson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Horn, P., Purcilly, A.: Routing number of dense and expanding graphs. J. Comb. 11(2), 329–350 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Johannsen, D.: Personal communication

  12. Krivelevich, M., Sudakov, B.: Pseudo-random graphs. In: More Sets, Graphs and Numbers. A Salute to Vera Sós and András Hajnal, pp. 199-262. Springer, János Bolyai Mathematical Society, Berlin, Budapest (2006)

  13. Leighton, F.T., Maggs, B.M., Rao, S.B.: Packet routing and job-shop scheduling in \(O\) (congestion + dilation) steps. Combinatorica 14(2), 167–186 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Letzter, S., Pokrovskiy, A., Yepremyan, L.: Size-Ramsey numbers of powers of hypergraph trees and long subdivisions. arXiv Preprint at arXiv:2103.01942 (2021)

  15. Li, W.-T., Lu, L., Yang, Y.: Routing numbers of cycles, complete bipartite graphs, and hypercubes. SIAM J. Discret. Math. 24(4), 1482–1494 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Montgomery, R.: Hamiltonicity in random graphs is born resilient. J. Comb. Theory Ser. B 139, 316–341 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Montgomery, R.: Spanning trees in random graphs. Adv. Math. 356, 92 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, L.: Optimal bounds for matching routing on trees. SIAM J. Discret. Math. 12(1), 64–77 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Rajko Nenadov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nenadov, R. Routing Permutations on Spectral Expanders via Matchings. Combinatorica 43, 737–742 (2023).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification