Abstract:
In this dissertation, we first discuss four versions of Hamiltonicity in hypergraphs. We mainly study the existence problem of Hamiltonian decompositions of uniform hypergraphs based on two versions of Hamiltonian cycles, so called ``KK-definition" and ``WJ-definition". For KK-definition, we create a recursive construction of KK-Hamiltonian decomposition of complete 3-uniform hypergraphs.
Our construction method uses a KK-Hamiltonian decomposition of the complete 3-uniform hypergraph, Kt(3), and some well-known graph decompositions to obtain a KK-Hamiltonian decomposition of the complete t-partite 3-uniform hypergraph, Kt(n)(3), when t=4,8 (mod12), n>=2, as well as a KK-Hamiltonian decomposition of K2t(3). Therefore, together with the current results in literatures, our method provides
a KK-Hamiltonian decomposition of the complete 3-uniform hypergraph, Kt(3), and the complete t-partite 3-uniform hypergraph, Kt(n)(3), when t=2m, 5*2m, 7*2m, 11*2m and m>=2, and n>=2. Furthermore, we establish a WJ-Hamiltonian decomposition of the complete 4-uniform bipartite hypergraph, Kn,n(4), where n=1(mod 4) and n is a prime number.
