A note on the characterization of digraphic sequences

Given pairs (a1,b1),…,(an,bn) of nonnegative integers, the digraph realization problem for digraphs asks whether there is a simple digraph (no loops or multiple arcs) with vertices v1,…,vn such that each vertex vi∈V has indegree ai and outdegree bi. Fulkerson and Chen obtained a characterization analogous to the classical Erdős-Gallai characterization for graphs, but with the additional constraint that the pairs must be sorted in nonincreasing lexicographical order. We provide a more general characterization that avoids the additional sorting. The inequalities needed correspond to those k such that ak+1>ak. We prove a similar result when one loop is allowed at each vertex.