Tournaments associated with multigraphs and a theorem of Hakimi

A tournament of order n is usually considered as an orientation of the complete graph K n . In this note, we consider a more general definition of a tournament that we call a C -tournament, where C is the adjacency matrix of a multigraph G , and a C -tournament is an orientation of G . The score vector of a C -tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C -tournament with a prescribed score vector R and gave an algorithm to construct such a C -tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimi's theorem, and then provide a direct construction of such a C -tournament which works even for weighted graphs.