Antidirected hamiltonian paths in $k$-hypertournaments
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| Format: | Preprint |
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2024
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| _version_ | 1866909229534674944 |
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| author | Yang, Hong Dong, Changchang Meng, Jixiang Liu, Juan |
| author_facet | Yang, Hong Dong, Changchang Meng, Jixiang Liu, Juan |
| contents | 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 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Antidirected hamiltonian paths in $k$-hypertournaments Yang, Hong Dong, Changchang Meng, Jixiang Liu, Juan Combinatorics 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 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). |
| title | Antidirected hamiltonian paths in $k$-hypertournaments |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.15851 |