Score lists in multipartite hypertournaments

TL;DR

This paper characterizes the score and losing score lists in multipartite hypertournaments, providing necessary and sufficient conditions for their realization based on combinatorial properties.

Contribution

It introduces a comprehensive framework for understanding score lists in $k$-partite hypertournaments, extending previous tournament theories to hypergraph structures.

Findings

Derived necessary and sufficient conditions for score lists.

Established criteria for losing score lists.

Extended tournament score theory to hypergraph structures.

Abstract

Given non-negative integers $n_{i}$ and $\alpha_{i}$ with $0 \leq \alpha_{i} \leq n_i$ $(i=1,2,...,k)$, an $[\alpha_{1},\alpha_{2},...,\alpha_{k}]$-$k$-partite hypertournament on $\sum_{1}^{k}n_{i}$ vertices is a $(k+1)$-tuple $(U_{1},U_{2},...,U_{k},E)$, where $U_{i}$ are $k$ vertex sets with $|U_{i}|=n_{i}$, and $E$ is a set of $\sum_{1}^{k}\alpha_{i}$-tuples of vertices, called arcs, with exactly $\alpha_{i}$ vertices from $U_{i}$, such that any $\sum_{1}^{k}\alpha_{i}$ subset $\cup_{1}^{k}U_{i}^{\prime}$ of $\cup_{1}^{k}U_{i}$, $E$ contains exactly one of the $(\sum_{1}^{k} \alpha_{i})!$ $\sum_{1}^{k}\alpha_{i}$-tuples whose entries belong to $\cup_{1}^{k}U_{i}^{\prime}$. We obtain necessary and sufficient conditions for $k$ lists of non-negative integers in non-decreasing order to be the losing score lists and to be the score lists of some $k$-partite hypertournament.