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Hypergraph expanders from Cayley graphs

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Abstract

We present a simple mechanism, which can be randomised, for constructing sparse 3-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over \(\mathbb{Z}_{2}^{t}\) and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.

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References

  1. N. Alon and V. D. Milman, λ 1, isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory. Series B 38 (1985), 73–88.

    Article  MathSciNet  Google Scholar 

  2. N. Alon and Y. Roichman, Random Cayley graphs and expanders, Random Structures & Algorithms 5 (1994), 271–284.

    Article  MathSciNet  Google Scholar 

  3. I. Bárány, A generalization of Carathéodory’s theorem, Discrete Mathematics 40 (1982), 141–152.

    Article  MathSciNet  Google Scholar 

  4. E. Boros and Z. Füredi, The number of triangles covering the center of an n-set, Geometiae Dedicata 17 (1984), 69–77.

    MathSciNet  MATH  Google Scholar 

  5. E. Breuillard, B. Green, R. Guralnick and T. Tao, Expansion in finite simple groups of Lie type, Journal of the European Mathematical Society 17 (2015), 1367–1434.

    Article  MathSciNet  Google Scholar 

  6. B. Bukh, A point in many triangles, Electronic Journal of Combinatorics 13 (2006), Note 10.

  7. B. Bukh, J. Matouček and G. Nivasch, Stabbing simplices by points and flats, Discrete & Computational Geometry 43 (2010), 321–338.

    Article  MathSciNet  Google Scholar 

  8. D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete Mathematics 185 (1998), 245–248.

    Article  MathSciNet  Google Scholar 

  9. D. Conlon, J. Tidor and Y. Zhao, Hypergraph expanders of all uniformities from Cayley graphs, arXiv:1809.06342.

  10. M. Desai and V. Rao, A characterization of the smallest eigenvalue of a graph, Journal of Graph Theory 18 (1994), 181–194.

    Article  MathSciNet  Google Scholar 

  11. J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Transactions of the American Mathematical Society 284 (1984), 787–794.

    Article  MathSciNet  Google Scholar 

  12. D. Dotterrer and M. Kahle, Coboundary expanders, Journal of Topology and Analysis 4 (2012), 499–514.

    Article  MathSciNet  Google Scholar 

  13. D. Dotterrer, T. Kaufman and U. Wagner, On expansion and topological overlap, in 32nd International Symposium on Computational Geometry, LIPIcs. Leibniz International Proceedings in Informatics, Vol. 51, Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern, 2016, Article 35.

    Google Scholar 

  14. A. Dudek, S. La Fleur, D. Mubayi and V. Roödl, On the size-Ramsey number of hyper-graphs, Journal of Graph Theory 86 (2017), 104–121.

    Article  MathSciNet  Google Scholar 

  15. S. Evra and T. Kaufman, Bounded degree cosystolic expanders of every dimension, in Proceedings of the 48th Annual ACM Symposium on Theory of Computing, ACM, New York, 2016, pp. 36–48.

    MATH  Google Scholar 

  16. J. Fox, M. Gromov, V. Lafforgue, A. Naor and J. Pach, Overlap properties of geometric expanders, Journal für die Reine und Angewandte Mathematik 671 (2012), 49–83.

    MathSciNet  MATH  Google Scholar 

  17. M. Gromov, Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry, Geometric and Functional Analysis 20 (2010), 416–526.

    Article  MathSciNet  Google Scholar 

  18. S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bulletin of the American Mathematical Society 43 (2006), 439–561.

    Article  MathSciNet  Google Scholar 

  19. T. Kaufman, D. Kazhdan and A. Lubotzky, Isoperimetric inequalities for Ramanujan complexes and topological expanders, Geometric and Functional Analysis 26 (2016), 250–287.

    Article  MathSciNet  Google Scholar 

  20. T. Kaufman and D. Mass, High dimensional random walks and colorful expansion, in 8th Innovations in Theoretical Computer Science Conference, LIPIcs. Leibniz International Proceedings in Informatics, Vol. 67, Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern, 2017, Article 4.

    Google Scholar 

  21. T. Kaufman and D. Mass, Walking on the edge and cosystolic expansion, arXiv:1606.01844.

  22. N. Linial and R. Meshulam, Homological connectivity of random 2-complexes, Combinatorica 26 (2006), 475–487.

    Article  MathSciNet  Google Scholar 

  23. L. Lu and Z. Wang, On the size-Ramsey number of tight paths, SIAM Journal on Discrete Mathematics 32 (2018), 2172–2179.

    Article  MathSciNet  Google Scholar 

  24. A. Lubotzky, Expander graphs in pure and applied mathematics, Bulletin of the American Mathematical Society 49 (2012), 113–162.

    Article  MathSciNet  Google Scholar 

  25. A. Lubotzky, Ramanujan complexes and high dimensional expanders, Japanese Journal of Mathematics 9 (2014), 137–169.

    Article  MathSciNet  Google Scholar 

  26. A. Lubotzky, Z. Luria and R. Rosenthal, Random Steiner systems and bounded degree coboundary expanders of every dimension, Discrete & Computational Geometry, to appear.

  27. A. Lubotzky and R. Meshulam, Random Latin squares and 2-dimensional expanders, Advances in Mathematics 272 (2015), 743–760.

    Article  MathSciNet  Google Scholar 

  28. A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.

    Article  MathSciNet  Google Scholar 

  29. A. Lubotzky, B. Samuels and U. Vishne, Ramanujan complexes of type à d, Israel Journal of Mathematics 149 (2005), 267–299.

    Article  MathSciNet  Google Scholar 

  30. G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problems of Information Transmission 24 (1988), 39–46.

    MathSciNet  MATH  Google Scholar 

  31. J. Pach, A Tverberg-type result on multicolored simplices, Computational Geometry 10 (1998), 71–76.

    Article  MathSciNet  Google Scholar 

  32. O. Parzanchevski, High dimensional expanders, PhD Thesis, Hebrew University, 2014.

  33. O. Parzanchevski, Mixing in high-dimensional expanders, Combinatorics, Probability and Computing 26 (2017), 746–761.

    Article  MathSciNet  Google Scholar 

  34. O. Parzanchevski, R. Rosenthal and R. Tessler, Isoperimetric inequalities in simplicial complexes, Combinatorica 36 (2016), 195–227.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author gratefully acknowledges the support of the Simons Institute for the Theory of Computing during part of the period when this paper was written. The author is also indebted to Noga Alon, who brought the problem of constructing high-dimensional expanders to his attention, and to Rajko Nenadov, Jonathan Tidor and Yufei Zhao for several valuable discussions.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    David Conlon

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  1. David Conlon
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Correspondence to David Conlon.

Additional information

Research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632.

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Conlon, D. Hypergraph expanders from Cayley graphs. Isr. J. Math. 233, 49–65 (2019). https://doi.org/10.1007/s11856-019-1895-1

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  • DOI: https://doi.org/10.1007/s11856-019-1895-1

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