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

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