The Generalization of Dirac's Theorem for Hypergraphs

A substantial amount of research in graph theory continues to concentrate on the existence of hamiltonian cycles and perfect matchings. A classic theorem of Dirac states that a sufficient condition for an n-vertex graph to be hamiltonian, and thus, for n even, to have a perfect matching, is that the minimum degree is at least n/2. Moreover, there are obvious counterexamples showing that this is best possible. The study of hamiltonian cycles in hypergraphs was initiated in [1] where, however, a different definition than the one considered here was introduced. Given an integer k ≥ 2, a k-uniform hypergraph is a hypergraph (a set system) where every edge (set) is of size k. By a cycle we mean a k-uniform hypergraph whose vertices can be ordered cyclically v1, . . . , vl in such a way that for each i = 1, . . . , l, the set {vi, vi+1, . . . , vi+k−1} is an edge, where for h > l we set vh = vh−l . A hamiltonian cycle in a k-uniform hypergraph H is a spanning cycle in H , that is, a sub-hypergraph of H which is a cycle and contains all vertices of H . A k-uniform hypergraph containing a hamiltonian cycle is called hamiltonian. This notion and its generalizations have a potential to be applicable in many contexts which still need to be explored. An application in the relational database theory can be found in [2]. As observed in [5], the square of a (graph) hamiltonian cycle naturally coincides with a hamiltonian cycle in a hypergraph built on top of the triangles of the graph. More precisely, given a graph G, let Tr(G) be the set of triangles in G. Define a hypergraph H(G) = (V (G), T r(G)). Then there is a one-to-one correspondence between hamiltonian cycles in H(G) and the squares of hamiltonian cycles in G. For results about the existence of squares of hamiltonian cycles see, e.g., [6]. As another potential application consider a seriously ill patient taking 24 different pills on a daily basis, one at a time every hour. Certain combinations