Opuscula Math. 40, no. 4 (2020), 509-516
https://doi.org/10.7494/OpMath.2020.40.4.509

Opuscula Mathematica

# Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length

R. Lakshmi
T. Poovaragavan

Abstract. A complete $$3$$-uniform hypergraph of order $$n$$ has vertex set $$V$$ with $$|V|=n$$ and the set of all $$3$$-subsets of $$V$$ as its edge set. A $$t$$-cycle in this hypergraph is $$v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1$$ where $$v_1, v_2,\dots, v_t$$ are distinct vertices and $$e_1, e_2,\dots, e_t$$ are distinct edges such that $$v_i, v_{i+1}\in e_i$$ for $$i \in \{1, 2,\dots, t-1\}$$ and $$v_t, v_1 \in e_t$$. A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete $$3$$-uniform hypergraph of order $$n$$ into $$p$$-cycles, whenever $$p$$ is prime.

Keywords: uniform hypergraph, cycle decomposition.

Mathematics Subject Classification: 05C65, 05C85.

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1. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1979.
2. J.C. Bermond, Hamiltonian decompositions of graphs, directed graphs and hypergraphs, Ann. Discrete Math. 3 (1978), 21-28.
3. J.C. Bermond, A. Germa, M.C. Heydemann, D. Sotteau, Hypergraphes hamiltoniens, [in:] Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, 39-43.
4. D. Bryant, S. Herke, B. Maenhaut, W. Wannasit, Decompositions of complete $$3$$-uniform hypergraphs into small $$3$$-uniform hypergraphs, Australas. J. Combin. 60 (2014) 2, 227-254.
5. H. Jordon, G. Newkirk, $$4$$-cycle decompositions of complete $$3$$-uniform hypergraphs, Australas. J. Combin. 71 (2018) 2, 312-323.
6. D. Kühn, D. Osthus, Decompositions of complete uniform hypergraphs into Hamilton Berge cycles, J. Combin. Theory Ser. A 126 (2014), 128-135.
7. R. Lakshmi, T. Poovaragavan, $$6$$-Cycle decompositions of complete $$3$$-uniform hypergraphs, (submitted).
8. P. Petecki, On cyclic hamiltonian decompositions of complete $$k$$-uniform hypergraphs, Discrete Math. 325 (2014), 74-76.
9. M. Truszczyński, Note on the decomposition of $$\lambda K_{m,n} (\lambda K_{m,n}^{*})$$ into paths, Discrete Math. 55 (1985), 89-96.
10. H. Verrall, Hamilton decompositions of complete $$3$$-uniform hypergraphs, Discrete Math. 132 (1994), 333-348.
• R. Lakshmi (corresponding author)
• https://orcid.org/0000-0001-9633-7676
• Annamalai University, Department of Mathematics, Annamalainagar-608 002, India
• Dharumapuram Gnanambigai Government Arts College for Women, Department of Mathematics, Mayiladuthurai-609 001, India
• Communicated by Andrzej Żak.
• Accepted: 2020-06-05.
• Published online: 2020-07-09.