The 3-uniform cycle of length 5 has five vertices a, b, c, d, e and five 3element edges abc, bcd, cde, dea, eab. Similarly, an r-uniform k-cycle has k vertices arranged in a cyclic order, and k edges which are the r-element subsets formed by any r consecutive vertices. A hypercycle system C(r, k, v) of order v is a collection of r-uniform k-cycles on a velement vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. In this paper we study hypercycle systems with r = 3 and k = 5. The definition of 2-split system is introduced, and recursive constructions of hypercycle systems C(3, 5, v) are designed. We find, by a new difference method, hypercycle systems C(3, 5, v) of orders v = 10, 11, 16, 20 and 22. By recursion, they yield infinite families of hypercycle systems.
|Number of pages||19|
|Journal||Australasian Journal of Combinatorics|
|Publication status||Published - 2020|
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics