HAMILTONIAN CYCLES IN TOURNAMENTS

Let V be a n-set (set of size n). Let E be the collection of all possible k-subsets (subsets of size k) of V (2 k n) each taken in one of its k! possible permutations. A pair T = (V; E) is called a hypertournament, or a k-tournament. Each element of V is a vertex, and each ordered k-tuple of E is a hyperedge or, simply, an edge. For vertices u; v 2 V and an edge e = (x1; : : : ; xk) 2 E, we say u dominates v via edge e if u precedes v in e, that is if u = xi, v = xj, 1 i < j k. We denote this by uev. A path consists of an alternating sequence x0e1x1e2x2: : : xk1elxl of distinct vertices xi and distinct edges ei so that xi1 dominates xi via ei (i = 1; : : : ; l). Such a path has length l. A cycle is a path when all vertices are distinct except x0 = xl. A path (cycle) of T is Hamiltonian if it contains all vertices of T. A k-tournament T is pancyclic if it con-tains cycles of all possible lengths. It is vertex-pancyclic if each vertex of T is contained in cycles of all possible lengths. A k-tournament