Fulkerson–Chen–Anstee theorem

Statement

The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers $((a_1,b_1),\ldots,(a_n,b_n))$ to be the indegree-outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson (1960) obtained a characterization analogous to the classical Erdős–Gallai theorem for graphs, but in contrast to this solution with exponentially many inequalities. In 1966 Chen improved this result in demanding the additional constraint that the integer pairs must be sorted in non-increasing lexicographical order leading to n inequalities. Anstee (1982) observed in a different context that it is sufficient to have $a_1\geq\cdots\geq a_n$. Berger reinvented this result and gives a direct proof.