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

Anita Keszler,Zsolt Tuza

doi:10.3390/math9050484

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.

Highlights

A hypergraph H of order v is a pair (X, E), where X is the vertex set with X = v and E is a family of subsets of X called edges
We deal with an object called the tight r-uniform hypercycle of length k (k > r ≥ 3)— called cycloid in another context—which consists of k vertices and k edges; namely, it is a cyclic sequence of k vertices of X in which any r consecutive vertices, and only those, form an edge
If a system is of even order 2v and contains two vertex-disjoint subsystems of order v, we say with the terminology of [9] that it is a 2-split system

Summary

Introduction

A hypergraph H of order v is a pair (X, E), where X is the vertex set with X = v and E is a family of subsets of X called edges. The complete r-uniform hypergraph of order v, denoted by Kv(r), is the hypergraph in which E consists of all the r-element subsets of X. The cyclic sequence (0, 1, 2, 3, 4) represents a C(3, 5) cycle for which the edges are the 3-sets {0, 1, 2}, {1, 2, 3}, {2, 3, 4}, {3, 4, 0}, and {4, 0, 1}. (Inside a 3-set, the order of vertices does not matter, but the order in a 5-tuple is of essence, except that any cyclic shift and the reversal of the sequence practically mean the same cycle.) The cyclic sequence (0, 1, 2, 3, 4) represents a C(3, 5) cycle for which the edges are the 3-sets {0, 1, 2}, {1, 2, 3}, {2, 3, 4}, {3, 4, 0}, and {4, 0, 1}. (Inside a 3-set, the order of vertices does not matter, but the order in a 5-tuple is of essence, except that any cyclic shift and the reversal of the sequence practically mean the same cycle.)

Objectives

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Conclusion