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
In a directed multigraph, the imbalance of a vertex vi​ is defined as
bvi​​=dvi​+​−dvi​−​, where dvi​+​ and
dvi​−​ denote the outdegree and indegree respectively of vi​. We
characterize imbalances in directed multigraphs and obtain lower and upper
bounds on imbalances in such digraphs. Also, we show the existence of a
directed multigraph with a given imbalance set