New proof of a Theorem on k-hypertournament losing scores
In this paper, we give a new and short proof of a Theorem on k-hypertournament losing scores due to Zhou et al.[7].
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arXiv subjects
S. Pirzada
Publications and source records attributed to S. Pirzada.
Score lists in [h-k]-bipartite hypertournaments
Given non-negative integers m, n, h and k with $ m\geq h>1 $ and $ n\geq k>1, $an [h-k]-bipartite hypertournament on $ m+n$ vertices is a triple $(U,V,A) $, where U and V are two sets of vertices with $| U| =m$ and $ | V| =n,$ and $A$ 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 any $h+k$ subsets $ U_{1}\cup V_{1}$ of $U\cup V, A$ contains exactly one of the $(h+k) ! (h+k) -$tuples whose entries belong to $U_{1}\cup V_{1}.$ We obtain necessary and sufficient conditions for a pair of non-decreasing sequences of non-negative integers to be the losing score lists or score lists of some$[h-k]-$bipartite hypertournament.\bigskip
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Score sets in oriented bipartite graphs
The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture when $\mid A\mid $ = 1, 2, 3, or when A is a geometric or arithmetic progression.
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