27 research outputs found
Reconstruction of complete interval tournaments
Let and be nonnegative integers ,
be a multigraph on vertices in which any pair of
vertices is connected with at least and at most edges and \textbf{v =}
be a vector containing nonnegative integers. We give
a necessary and sufficient condition for the existence of such orientation of
the edges of , that the resulted out-degree vector equals
to \textbf{v}. We describe a reconstruction algorithm. In worst case checking
of \textbf{v} requires time and the reconstruction algorithm works
in time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the
score sequences of tournaments are special cases resp. of our result
On Scores, Losing Scores and Total Scores in Hypertournaments
A -hypertournament is a complete -hypergraph with each -edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a -hypertournament, the score (losing score ) of a vertex is the number of arcs containing in which is not the last element (in which is the last element). The total score of is defined as . In this paper we obtain stronger inequalities for the quantities , and , where . Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong -hypertournaments
Imbalances in directed multigraphs
In a directed multigraph, the imbalance of a vertex is defined as
, where and
denote the outdegree and indegree respectively of . 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
Score lists in multipartite hypertournaments
Given non-negative integers and with , an
--partite hypertournament on
vertices is a -tuple ,
where are vertex sets with , and is a set of
-tuples of vertices, called arcs, with exactly
vertices from , such that any
subset of , contains
exactly one of the -tuples
whose entries belong to . We obtain necessary and
sufficient conditions for lists of non-negative integers in non-decreasing
order to be the losing score lists and to be the score lists of some
-partite hypertournament