Given non-negative integers ni​ and αi​ with 0≤αi​≤ni​(i=1,2,...,k), an
[α1​,α2​,...,αk​]-k-partite hypertournament on
∑1k​ni​ vertices is a (k+1)-tuple (U1​,U2​,...,Uk​,E),
where Ui​ are k vertex sets with ∣Ui​∣=ni​, and E is a set of
∑1k​αi​-tuples of vertices, called arcs, with exactly
αi​ vertices from Ui​, such that any ∑1k​αi​
subset ∪1k​Ui′​ of ∪1k​Ui​, E contains
exactly one of the (∑1k​αi​)!∑1k​αi​-tuples
whose entries belong to ∪1k​Ui′​. 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
A k-hypertournament is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score si​ (losing score ri​) of a vertex vi​ is the number of arcs containing vi​ in which vi​ is not the last element (in which vi​ is the last element). The total score of vi​ is defined as ti​=si​−ri​. In this paper we obtain stronger inequalities for the quantities ∑i∈I​ri​, ∑i∈I​si​ and ∑i∈I​ti​, where I⊆{1,2,…,n}. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong k-hypertournaments
An open problem posed by the first author is the complexity to decide whether
a sequence of nonnegative integer numbers can be the final score of a football
tournament. In this paper we propose polynomial time approximate and
exponential time exact algorithms which solve the problem