Antidirected hamiltonian paths in $k$-hypertournaments

Abstract

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 $x_1 a_1 x_2 a_2 x_3 \ldots x_{t-1} a_{t-1} x_t$ 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 $a_{i}$. 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 $2\leq k\leq n-1$, has an antidirected hamiltonian path, which extends Gr\"{u}nbaum'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).