A k-hypertournament H on n vertices is a pair (V(H), A(H)), where V(H) is a set of vertices and A(H) is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V(H), A(H) contains exactly one of the k! k-tuples whose entries belong to S. Clearly, a 2-hypertournament is a tournament. An antidirected path in H is a sequence \(x_1 a_1 x_2 a_2 x_3 \cdots x_{t-1} a_{t-1} x_t\) of distinct vertices \(x_1, x_2, \ldots , x_t\) and distinct arcs \(a_1, a_{2},\ldots , a_{t-1}\) such that for any \(i\in \{2,3,\ldots , t-1\}\) , either \(x_{i-1}\) precedes \(x_{i}\) in \(a_{i-1}\) and \(x_{i+1}\) precedes \(x_{i}\) in \(a_{i}\) , or \(x_{i}\) precedes \(x_{i-1}\) in \(a_{i-1}\) and \(x_{i}\) precedes \(x_{i+1}\) in \(a_{i}\) , but not both. An antidirected path that includes all vertices of a k-hypertournament H is known as an antidirected Hamilton path of H. In this paper, we prove that except for four k-hypertournaments, \(T_3^{c}, T_5^{c}, T_7^{c}\) and \(H_{4}\) , every k-hypertournament with n vertices, where \(2\le k\le n-1\) , has an antidirected Hamilton path, which extends Grünbaum’s theorem on tournaments.
