Abstract

Given non-negative integers $m$, $n$, $h$ and $k$ with $m\ge h\ge 1$ and $n\ge k\ge 1$, an $[h,k]$-bipartite multi hypertournament (or briefly $[h,k]$-BMHT) on $m+n$ vertices is a triple $(U,V,\bold A)$, where $U$ and $V$ are two sets of vertices with $|U|=m$ and $|V|=n$ and $\bold A$ is a set of $(h+k$)-tuples of vertices, called arcs with exactly $h$ vertices from $U$ and exactly $k$ vertices from $V$, such that for any $h+k$ subset $U_{1}\cup V_{1}$ of $U\cup V$, $\bold A$ contains at least one and at most $(h+k)!$ $(h+k)$-tuples whose entries belong to $U_{1}\cup V_{1}$. If $\bold A$ is a set of $(r+s)$-tuples of vertices, called arcs for $r$ ($1\leq r\leq h$) vertices from $U$ and $s$ ($1\leq s\leq k$) vertices from $V$ such that $\bold A$ contains at least one and at most $(r+s)!$ $(r+s)$-tuples, then the bipartite multi hypertournament is called an $(h,k)$-bipartite multi hypertournament (or briefly $(h,k)$-BMHT). We obtain necessary and sufficient conditions for a pair of sequences of non-negative integers in non-decreasing order to be losing score lists and score lists of $[h,k]$-BMHT and $(h,k)$-BMHT.

Keywords: Hypertournaments; bipartite hypertournaments; score; losing score.

MSC: 05C65

Pages:  286--296     

Volume  64 ,  Issue  4 ,  2012