Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T20:24:27.993Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimum Codegree Threshold for C63-Factors in 3-Uniform Hypergraphs

Published online by Cambridge University Press:  22 March 2017

WEI GAO  and
JIE HAN
WEI GAO
Affiliation:
Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA (e-mail: wzg0021@auburn.edu)
JIE HAN
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, Brazil (e-mail: jhan@ime.usp.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let C63 be the 3-uniform hypergraph on {1, . . ., 6} with edges 123,345,561, which can be seen as the analogue of the triangle in 3-uniform hypergraphs. For sufficiently large n divisible by 6, we show that every n-vertex 3-uniform hypergraph H with minimum codegree at least n/3 contains a C63-factor, that is, a spanning subhypergraph consisting of vertex-disjoint copies of C63. The minimum codegree condition is best possible. This improves the asymptotic result obtained by Mycroft and answers a question of Rödl and Ruciński exactly.

Keywords

Primary 05C70Secondary 05C65
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 536 - 559
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alon, N., Frankl, P., Huang, H., Rödl, V., Ruciński, A. and Sudakov, B. (2012) Large matchings in uniform hypergraphs and the conjecture of Erdős and Samuels. J. Combin. Theory Ser. A 119 1200–1215.Google Scholar
[2] Alon, N. and Yuster, R. (1996) H-factors in dense graphs. J. Combin. Theory Ser. B 66 269–282.Google Scholar
[3] Corradi, K. and Hajnal, A. (1963) On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar. 14 423–439.CrossRefGoogle Scholar
[4] Czygrinow, A. (2016) Tight co-degree condition for packing of loose cycles in 3-graphs. J. Graph Theory 83 317–333.Google Scholar
[5] Czygrinow, A. Minimum degree condition for C 4-tiling in 3-uniform hypergraphs. Preprint.Google Scholar
[6] Czygrinow, A., DeBiasio, L. and Nagle, B. (2014) Tiling 3-uniform hypergraphs with K 4 3-2e. J. Graph Theory 75 124–136.Google Scholar
[7] Czygrinow, A. and Kamat, V. (2012) Tight co-degree condition for perfect matchings in 4-graphs. Electron. J. Combin. 19 #P20.CrossRefGoogle Scholar
[8] Czygrinow, A. and Molla, T. (2014) Tight codegree condition for the existence of loose Hamilton cycles in 3-graphs. SIAM J. Discrete Math. 28 67–76.Google Scholar
[9] Erdős, P. (1964) On extremal problems of graphs and generalized graphs. Israel J. Math. 2 183–190.Google Scholar
[10] Gao, W., Han, J. and Zhao, Y. Codegree conditions for tiling complete k-partite k-graphs and loose cycles. arXiv:1612.07247 Google Scholar
[11] Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications II: Balatonfüred 1969, North-Holland, pp. 601–623.Google Scholar
[12] Hàn, H. and Schacht, M. (2010) Dirac-type results for loose Hamilton cycles in uniform hypergraphs. J. Combin. Theory Ser. B 100 332–346.Google Scholar
[13] Han, J. (2015) Near perfect matchings in k-uniform hypergraphs. Combin. Probab. Comput. 24 723–732.Google Scholar
[14] Han, J., Lo, A., Treglown, A. and Zhao, Y. Exact minimum codegree threshold for K 4 −-factors. Preprint.Google Scholar
[15] Han, J. and Treglown, A. The complexity of perfect matchings and packings in dense graphs and hypergraphs. Preprint.Google Scholar
[16] Han, J., Zang, C. and Zhao, Y. Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs. J. Combin. Theory Ser. A 149 115–147.Google Scholar
[17] Han, J. and Zhao, Y. (2015) Minimum degree thresholds for loose Hamilton cycle in 3-graphs. J. Combin. Theory Ser. B 114 70–96.CrossRefGoogle Scholar
[18] Han, J. and Zhao, Y. (2015) Minimum vertex degree threshold for C 4 3-tiling. J. Graph Theory 79 300–317.Google Scholar
[19] Hell, P. and Kirkpatrick, D. G. (1983) On the complexity of general graph factor problems. SIAM J. Comput. 12 601–609.Google Scholar
[20] Keevash, P. (2011) A hypergraph blow-up lemma. Random Struct. Alg. 39 275–376.Google Scholar
[21] Keevash, P. and Mycroft, R. (2014) A Geometric Theory for Hypergraph Matching, Vol. 233 of Memoirs of the American Mathematical Society, AMS.Google Scholar
[22] Khan, I. (2013) Perfect matchings in 3-uniform hypergraphs with large vertex degree. SIAM J. Discrete Math. 27 1021–1039.Google Scholar
[23] Khan, I. (2016) Perfect matchings in 4-uniform hypergraphs. J. Combin. Theory Ser. B 116 333–366.Google Scholar
[24] Komlós, J., Sárközy, G. and Szemerédi, E. (2001) Proof of the Alon–Yuster conjecture. Discrete Math. 235 255–269.Google Scholar
[25] Kühn, D. and Osthus, D. (2006) Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree. J. Combin. Theory Ser. B 96 767–821.Google Scholar
[26] Kühn, D. and Osthus, D. (2006) Multicolored Hamilton cycles and perfect matchings in pseudorandom graphs. SIAM J. Discrete Math. 20 273–286.Google Scholar
[27] Kühn, D. and Osthus, D. (2009) Embedding large subgraphs into dense graphs. In Surveys in Combinatorics 2009, Vol. 365 of London Math. Society Lecture Note Series, Cambridge University Press, pp. 137–167.Google Scholar
[28] Kühn, D. and Osthus, D. (2009) The minimum degree threshold for perfect graph packings. Combinatorica 29 65–107.Google Scholar
[29] Kühn, D., Osthus, D. and Treglown, A. (2013) Matchings in 3-uniform hypergraphs. J. Combin. Theory Ser. B 103 291–305.Google Scholar
[30] Lo, A. and Markström, K. (2013) Minimum codegree threshold for (K 4 3-e)-factors. J. Combin. Theory Ser. A 120 708–721.Google Scholar
[31] Lo, A. and Markström, K. (2015) F-factors in hypergraphs via absorption. Graphs Combin. 31 679–712.Google Scholar
[32] Mycroft, R. (2016) Packing k-partite k-uniform hypergraphs. J. Combin. Theory Ser. A 138 60–132.Google Scholar
[33] Rödl, V. and Ruciński, A. (2010) Dirac-type questions for hypergraphs: A survey (or more problems for Endre to solve). In An Irregular Mind: Szemerédi is 70, Vol. 21 of Bolyai Society Mathematical Studies, Springer, pp. 561–590.CrossRefGoogle Scholar
[34] Rödl, V., Ruciński, A. and Szemerédi, E. (2006) A Dirac-type theorem for 3-uniform hypergraphs. Combin. Probab. Comput. 15 229–251.Google Scholar
[35] 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
[36] Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes: Orsay 1976), Vol. 260 of Colloq. Internat. CNRS, CNRS, Paris, pp. 399–401.Google Scholar
[37] Treglown, A. and Zhao, Y. (2013) Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II. J. Combin. Theory Ser. A 120 1463–1482.Google Scholar
[38] Treglown, A. and Zhao, Y. (2016) A note on perfect matchings in uniform hypergraphs. Electron. J. Combin. 23 #P1.16.Google Scholar
[39] Tutte, W. T. (1947) The factorization of linear graphs. J. London Math. Soc. 22 107–111.Google Scholar
[40] Zhao, Y. (2015) Recent advances on Dirac-type problems for hypergraphs. In Recent Trends in Combinatorics, Vol. 159 of The IMA Volumes in Mathematics and its Applications, Springer.Google Scholar