Score lists in bipartite multi hypertournaments

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 $(U,V,\text{boldA})$, where $U$ and $V$ are two sets of vertices with $|U|=m$ and$|V|=n$ and $\mathbf{A}$ is a set of $(h+k$)-tuples of vertices, calledarcs with exactly $h$ vertices from $U$ and exactly $k$ verticesfrom $V$, such that for any $h+k$ subset $U_1\cup V_1$ of$U\cup V$, $\mathbf{A}$ contains at least one and at most $(h+k)!$$(h+k)$-tuples whose entries belong to $U_1\cup V_1$. If $\text{boldA}$ is a set of $(r+s)$-tuples of vertices, called arcs for $r$($1\le r\le h$) vertices from $U$ and $s$ ($1\le s\le k$)vertices from $V$ such that $\mathbf{A}$ contains at least one and atmost $(r+s)!$ $(r+s)$-tuples, then the bipartite multihypertournament is called an $(h,k)$-bipartite multihypertournament (or briefly $(h,k)$-BMHT). We obtain necessary andsufficient conditions for a pair of sequences of non-negativeintegers in non-decreasing order to be losing score lists andscore lists of $[h,k]$-BMHT and $(h,k)$-BMHT.