On Score Sequences ofk-Hypertournaments

Given two nonnegative integers n and k withn ≥ k > 1, a k -hypertournament on n vertices is a pair (V, A), where V is a set of vertices with | V | = n and A is a set of k -tuples of vertices, called arcs, such that for any k -subset S ofV , A contains exactly one of the k!k -tuples whose entries belong to S. We show that a nondecreasing sequence (r1, r2,⋯ , rn) of nonnegative integers is a losing score sequence of a k -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. We also show that a nondecreasing sequence (s1,s2 ,⋯ , sn) of nonnegative integers is a score sequence of somek -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. Furthermore, we obtain a necessary and sufficient condition for a score sequence of a strong k -hypertournament. The above results generalize the corresponding theorems on tournaments.