Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

Anita Keszler and Zsolt Tuza
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Anita Keszler: Machine Perception Laboratory, Institute for Computer Science and Control (SZTAKI), Kende u. 13-17, 1111 Budapest, Hungary
Zsolt Tuza: Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, 1053 Budapest, Hungary

Mathematics, 2021, vol. 9, issue 5, 1-59

Abstract: In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C ( r , k , v ) of order v is a collection of r -uniform k -cycles on a v -element vertex set, such that each r -element subset is an edge in precisely one of those k -cycles. We present cyclic hypercycle systems C ( 3 , 5 , v ) of orders v = 25 , 26 , 31 , 35 , 37 , 41 , 46 , 47 , 55 , 56 , a highly symmetric construction for v = 40 , and cyclic 2-split constructions of orders 32 , 40 , 50 , 52 . As a consequence, all orders v ? 60 permitted by the divisibility conditions admit a C ( 3 , 5 , v ) system. New recursive constructions are also introduced.

Keywords: hypergraph; hypercycle system; 3-uniform 5-cycle; edge decomposition; Steiner system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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