Degree frequencies in digraphs and tournaments

The number of vertices in a digraph G having a particular outdegree (indegree) is called the frequency of the outdegree (indegree). A set F of distinct positive integers {f₁, f₂, …, f_n} is the frequency set of the digraph G if every outdegree and indegree occurs with frequency f_j∈F and for each f_j∈F there is a least one outdegree and at least one indegree with frequency f_j. We prove that each nonempty set F of positive integers is the frequency set of some tournament, and we determine the smallest possible order for such a tournament. Similar results for asymmetric digraphs are also given. The results and techniques for frequency sets are used to derive corresponding results for vertex frequency partitions.