Antidirected hamiltonian paths in k-hypertournaments

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 x₁ a₁ x₂ a₂ x₃ … xₜ₋₁ aₜ₋₁ xₜ of distinct vertices x₁, x₂, …, xₜ and distinct arcs a₁, a₂,…, aₜ₋₁ such that for any i∈ \2,3,…, t-1\, either xᵢ₋₁ precedes xᵢ in aᵢ₋₁ and xᵢ₊₁ precedes xᵢ in aᵢ, or xᵢ precedes xᵢ₋₁ in aᵢ₋₁ and xᵢ precedes xᵢ₊₁ in aᵢ. An antidirected path that includes all vertices of H is known as an antidirected hamiltonian path. In this paper, we prove that except for four hypertournaments, T₃ᶜ, T₅ᶜ, T₇ᶜ and H₄, every k-hypertournament with n vetices, where 2≤ k≤ n-1, has an antidirected hamiltonian path, which extends Gr\"{u}nbaum's theorem on tournaments (except for three tournaments, T₃ᶜ, T₅ᶜ and T₇ᶜ, every tournament has an antidirected hamiltonian path).