Opuscula Mathematica
Opuscula Math. 40, no. 2 (2020), 209-225

Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles

Ryan C. Bunge
Dontez Collins
Daryl Conko-Camel
Saad I. El-Zanati
Rachel Liebrecht
Alexander Vasquez
Abstract. It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2\ (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges.
Keywords: maximum packing, \(\lambda\)-fold complete 3-uniform hypergraph, loose 3-cycle.
Mathematics Subject Classification: 05C65, 05C85.
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References ▼
  1. [1] P. Adams, D. Bryant, M. Buchanan, A survey on the existence of \(G\)-designs, J. Combin. Des. 16 (2008), 373-410.
  2. [2] R.F. Bailey, B. Stevens, Hamilton decompositions of complete \(k\)-uniform hypergraphs, Discrete Math. 310 (2010), 3088-3095.
  3. [3] Zs. Baranyai, On the factorization of the complete uniform hypergraph, [in:] Infinite and finite sets, Colloq. Math. Soc. János Bolyai, 10, North-Holland, Amsterdam, 1975, 91-108.
  4. [4] J.-C. Bermond, A. Germa, D. Sotteau, Hypergraph-designs, Ars Combinatoria 3 (1977), 47-66.
  5. [5] D.E. Bryant, T.A. McCourt, Existence results for \(G\)-designs, http://wiki.smp.uq.edu.au/G-designs/
  6. [6] D. Bryant, S. Herke, B. Maenhaut, W. Wannasit, Decompositions of complete 3-uniform hypergraphs into small 3-uniform hypergraphs, Australas. J. Combin. 60 (2014), 227-254.
  7. [7] R.C. Bunge, S.I. El-Zanati, L. Haman, C. Hatzer, K. Koe, K. Spornberger, On loose 4-cycle decompositions of complete 3-uniform hypergraphs, Bull. Inst. Combin. Appl. 87 (2019), 75-84.
  8. [8] C.J. Colbourn, R. Mathon, Steiner systems, [in:] The CRC Handbook of Combinatorial Designs, 2nd ed. (eds. C.J. Colbourn, J.H. Dinitz), CRC Press, Boca Raton (2007), 102-110.
  9. [9] S. Glock, D. Kühn, A. Lo, D. Osthus, The existence of designs via iterative absorption, arXiv:1611.06827v2 (2017).
  10. [10] S. Glock, D. Kühn, A. Lo, D. Osthus, Hypergraph \(F\)-designs for arbitrary \(F\), arXiv:1706.01800 (2017).
  11. [11] H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960), 145-157.
  12. [12] H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), pp. 260-264, Ann. New York Acad. Sci., vol. 319, New York Acad. Sci., New York, 1979.
  13. [13] H. Jordon, G. Newkirk, 4-cycle decompositions of complete 3-uniform hypergraphs, Australas. J. Combin. 71 (2018), 312-323.
  14. [14] P. Keevash, The existence of designs, arXiv:1401.3665v2 (2018).
  15. [15] G.B. Khosrovshahi, R. Laue, \(t\)-designs with \(t\geq 3\), [in:] The CRC Handbook of Combinatorial Designs, 2nd ed. (eds. C.J. Colbourn, J.H. Dinitz), CRC Press, Boca Raton (2007), 79-101.
  16. [16] J. Kuhl, M.W. Schroeder, Hamilton cycle decompositions of \(k\)-uniform \(k\)-partite hypergraphs, Australas. J. Combin. 56 (2013), 23-37.
  17. [17] D. Kühn, D. Osthus, Decompositions of complete uniform hypergraphs into Hamilton Berge cycles, J. Combin. Theory Ser. A 126 (2014), 128-135.
  18. [18] Z. Lonc, Solution of a delta-system decomposition problem, J. Combin. Theory, Ser. A 55 (1990), 33-48.
  19. [19] Z. Lonc, Packing, covering and decomposing of a complete uniform hypergraph into delta-systems, Graphs Combin. 8 (1992), 333-341.
  20. [20] M. Meszka, A. Rosa, Decomposing complete \(3\)-uniform hypergraphs into Hamiltonian cycles, Australas. J. Combin. 45 (2009), 291-302.
  21. [21] A.-F. Mouyart, F. Sterboul, Decomposition of the complete hypergraph into delta-systems II, J. Combin. Theory, Ser. A 41 (1986), 139-149.
  22. [22] M.W. Schroeder, On Hamilton cycle decompositions of \(r\)-uniform \(r\)-partite hypergraphs, Discrete Math. 315 (2014), 1-8.
  23. [23] R.M. Wilson, Decompositions of Complete Graphs into Subgraphs Isomorphic to a Given Graph, [in:] Proc. Fifth British Combinatorial Conference (eds. C.St.J.A. Nash-Williams, J. Sheehan), pp. 647-659, Congr. Numer. XV, 1975.
About the authors ▼
Ryan C. Bunge
https://orcid.org/0000-0003-0051-379X
Illinois State University, Normal, IL 61790, USA
Dontez Collins
https://orcid.org/0000-0003-4009-9642
Sussex Technical High School, Georgetown, DE 19947, USA
Daryl Conko-Camel
https://orcid.org/0000-0002-0982-5565
Salish Kootenai College, Pablo, MT 59855, USA
Saad I. El-Zanati (corresponding author)
https://orcid.org/0000-0002-7680-5433
Illinois State University, Normal, IL 61790, USA
Rachel Liebrecht
https://orcid.org/0000-0002-2635-2210
Ohio Northern University, Ada, OH 45810, USA
Alexander Vasquez
https://orcid.org/0000-0001-5608-7553
Manhattan College, Bronx, NY 10471, USA
About the article ▼
Communicated by Adam Paweł Wojda.
Received: 2019-08-28.
Revised: 2020-01-23.
Accepted: 2020-01-23.
Published online: 2020-03-09.
Cite this article as ▼
Ryan C. Bunge, Dontez Collins, Daryl Conko-Camel, Saad I. El-Zanati, Rachel Liebrecht, Alexander Vasquez, Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles, Opuscula Math. 40, no. 2 (2020), 209-225, https://doi.org/10.7494/OpMath.2020.40.2.209.
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