Score lists in multipartite hypertournaments

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.