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

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.