On Hamiltonian decompositions of complete 3-uniform hypergraphs

Abstract

Based on the definition of Hamiltonian cycles by Katona and Kierstead, we present a recursive construction of tight Hamiltonian decompositions of complete 3-uniform hypergraphs $K_n^{\left(3\right)}$, and complete multipartite 3-uniform hypergraph $K_{t\left(n\right)}^{\left(3\right)}$, where t is the number of partite sets and n is the size of each partite set. For $t\equiv 4,8\left(\mathrm{mod}12\right)$, we utilize a tight Hamiltonian decomposition of $K_t^{\left(3\right)}$ to construct those of $K_{2t}^{\left(3\right)}$ and $K_{t\left(n\right)}^{\left(3\right)}$ for all positive integers n. By applying our method in conjunction with the current results in literature, we obtain tight Hamiltonian decompositions for infinitely many hypergraphs, namely complete hypergraphs $K_t^{\left(3\right)}$ and complete multipartite hypergraphs $K_{t\left(n\right)}^{\left(3\right)}$ for any positive integer n, and $t=2^m,5\cdot 2^m,7\cdot 2^m$, and $11\cdot 2^m$ when $m\ge 2$.