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 $x_1{a}_1{x}_2{a}_2{x}_3\dots x_{t-1}{a}_{t-1}{x}_t$ of distinct vertices $x_1,x_2,\dots ,x_t$ and distinct arcs $a_1,a_2,\dots ,a_{t-1}$ such that for any $i\in \{2,3,\dots ,t-1\}$, 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 $a_i$. 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, $T_3^c,T_5^c,T_7^c$ and $H_4$, every $k$-hypertournament with $n$ vetices, where $2\le k\le n-1$, has an antidirected hamiltonian path, which extends Grünbaum’s theorem on tournaments (except for three tournaments, $T_3^c,T_5^c$ and $T_7^c$, every tournament has an antidirected hamiltonian path).