---
tag: T78S3AWT0R
kind: statement
role: proposition
name: Fulkerson–Chen–Anstee theorem
sources:
  - wikipedia: en:43198751@r1320899116
    locator: Fulkerson–Chen–Anstee theorem
    note: adopted
    mention: wikipedia://en/43198751
    sections: [Statement]
same_as:
  wikidata: Q18206243
  wikipedia: Fulkerson–Chen–Anstee theorem
inventory: [ETZ00NGBM1]
---

## Statement

The **Fulkerson–Chen–Anstee theorem** is a result in graph theory, a branch of [[combinatorics|0JKW7W7XFR]]. 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|0C4JS9RF5M]] $((a_1,b_1),\ldots,(a_n,b_n))$ to be the [[indegree|0KFYGSHHF5]]-[[outdegree|0KFYGSHHF5]] pairs of a [[simple directed graph|0KFYGSHHF5]]; 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|0B68KNPGCG]] 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.