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Pseudorandom hypergraph matchings

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  22 July 2020

Stefan Ehard ,
Stefan Glock  and
Felix Joos
Stefan Ehard
Affiliation:
Institut für Optimierung und Operations Research, Universität Ulm, Germany
Stefan Glock*
Affiliation:
Institute for Theoretical Studies, ETH Zürich, Switzerland
Felix Joos
Affiliation:
Institut für Informatik, Universität Heidelberg, Germany
*
*Corresponding author. Email: stefan.glock@eth-its.ethz.ch
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Abstract

A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs 05C65: Hypergraphs 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05D15: Transversal (matching) theory 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 868 - 885
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

The research leading to these results was partially supported by the EPSRC, grants EP/N019504/1 (S. Glock) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 339933727 (F. Joos).

References

Aharoni, R., Georgakopoulos, A. and Sprüssel, P. (2009) Perfect matchings in r-partite r-graphs. European J. Combin. 30 39–42.CrossRefGoogle Scholar
Ajtai, M., Komlós, J. and Szemerédi, E. (1981) A dense infinite Sidon sequence. European J. Combin. 2 1–11.CrossRefGoogle Scholar
Alon, N. and Yuster, R. (2005) On a hypergraph matching problem. Graphs Combin. 21 377–384.CrossRefGoogle Scholar
Condon, P., Kim, J., Kühn, D. and Osthus, D. (2019) A bandwidth theorem for approximate decompositions. Proc. Lond. Math. Soc. 118 1393–1449.CrossRefGoogle Scholar
Cooper, C., Frieze, A., Molloy, M. and Reed, B. (1996) Perfect matchings in random r-regular, s-uniform hypergraphs. Combin. Probab. Comput. 5 1–14.CrossRefGoogle Scholar
Ehard, S., Glock, S. and Joos, F. (2019) A rainbow blow-up lemma for almost optimally bounded edge-colourings.arXiv:1907.09950Google Scholar
Ehard, S. and Joos, F. (2020) A short proof of the blow-up lemma for approximate decompositions.arXiv:2001.03506Google Scholar
Frankl, P. and Rödl, V. (1985) Near perfect coverings in graphs and hypergraphs. European J. Combin. 6 317–326.CrossRefGoogle Scholar
Freedman, D. A. (1975) Packing Hamilton cycles in random and pseudo-random hypergraphs. Random Struct. Algorithms 41 1–22.Google Scholar
Frieze, A. and Krivelevich, K. (2012) Packing Hamilton cycles in random and pseudo-random hypergraphs. Random Struct. Algorithms 41 1–22.CrossRefGoogle Scholar
Füredi, Z. (1988) Matchings and covers in hypergraphs. Graphs Combin. 4 115–206.CrossRefGoogle Scholar
Glock, S., Kühn, D., Lo, A. and Osthus, D. The existence of designs via iterative absorption: hypergraph F-designs for arbitrary F. Mem. Amer. Math. Soc., to appear.Google Scholar
Hàn, H., Han, J. and Morris, P. (2020) Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs. In 14th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 702–717, SIAM.CrossRefGoogle Scholar
Hàn, H., Person, Y. and Schacht, M. (2009) On perfect matchings in uniform hypergraphs with large minimum vertex degree. SIAM J. Discrete Math. 23 732–748.CrossRefGoogle Scholar
Johansson, A., Kahn, J. and Vu, V. (2008) Factors in random graphs. Random Struct. Algorithms 33 1–28.CrossRefGoogle Scholar
Joos, F., Kim, J., Kühn, D. and Osthus, D. (2019) Optimal packings of bounded degree trees. J. Eur. Math. Soc. 21 3573–3647.CrossRefGoogle Scholar
Kahn, J. (1996) Asymptotically good list-colorings. J. Combin. Theory Ser. A 73 1–59.CrossRefGoogle Scholar
Kahn, J. (1996) A linear programming perspective on the Frankl–Rödl–Pippenger theorem. Random Struct. Algorithms 8 149–157.3.0.CO;2-Y>CrossRefGoogle Scholar
Karp, R. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations (Miller, R. E. et al., eds), pp. 85–103, Springer.CrossRefGoogle Scholar
Keevash, P. (2014) The existence of designs.arXiv:1401.3665Google Scholar
Keevash, P. (2018) The existence of designs II.arXiv:1802.05900Google Scholar
Keevash, P. (2018) Hypergraph matchings and designs. Proc. Int. Cong. Math. 3 3099–3122.Google Scholar
Keevash, P. and Mycroft, R. (2015) A Geometric Theory for Hypergraph Matching, Vol. 233, no. 1098 of Memoirs of the American Mathematical Society, AMS.Google Scholar
Kim, J., Kühn, D., Osthus, D. and Tyomkyn, M. (2019) A blow-up lemma for approximate decompositions. Trans. Amer. Math. Soc. 371 4655–4742.CrossRefGoogle Scholar
Klenke, A. (2014) Probability Theory, second edition, Springer.CrossRefGoogle Scholar
Komlós, J., Pintz, J. and Szemerédi, E. (1982) A lower bound for Heilbronn’s problem. J. London Math. Soc. (2) 25 13–24.CrossRefGoogle Scholar
Komlós, J., Sárközy, G. N. and Szemerédi, E. (1997) Blow-up lemma. Combinatorica 17 109–123.CrossRefGoogle Scholar
Lenz, J. and Mubayi, D. (2016) Perfect packings in quasirandom hypergraphs I. J. Combin. Theory Ser. B 119 155–177.CrossRefGoogle Scholar
McDiarmid, C. (1989) On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989), Vol. 141 of London Mathematical Society Lecture Note Series, pp. 148–188, Cambridge University Press.CrossRefGoogle Scholar
Molloy, M. and Reed, B. (2000) Near-optimal list colorings. Random Struct. Algorithms 17 376–402.3.0.CO;2-0>CrossRefGoogle Scholar
Montgomery, R., Pokrovskiy, A. and Sudakov, B. (2019) Decompositions into spanning rainbow structures. Proc. Lond. Math. Soc. 119 899–959.CrossRefGoogle Scholar
Montgomery, R., Pokrovskiy, A. and Sudakov, B. (2020) A proof of Ringel’s conjecture.arXiv:2001.02665Google Scholar
Montgomery, R., Pokrovskiy, A. and Sudakov, B. Embedding rainbow trees with applications to graph labelling and decomposition. J. Eur. Math. Soc., to appear.Google Scholar
Pippenger, N. and Spencer, J. (1989) Asymptotic behaviour of the chromatic index for hypergraphs. J. Combin. Theory Ser. A 51 24–42.CrossRefGoogle Scholar
Rödl, V. (1985) On a packing and covering problem. European J. Combin. 6 69–78.CrossRefGoogle Scholar
Rödl, V. and Ruciński, A. (2010) Dirac-type questions for hypergraphs: a survey (or more problems for Endre to solve). In An Irregular Mind (Bárány, I. and Solymosi, J., eds), Vol. 21 of Bolyai Society Mathematical Studies, pp. 561–590, János Bolyai Mathematical Society.CrossRefGoogle Scholar
Rödl, V., Ruciński, A. and Szemerédi, E. (2009) Perfect matchings in large uniform hypergraphs with large minimum collective degree. J. Combin. Theory Ser. A 116 613–636.CrossRefGoogle Scholar