# Fulkerson–Chen–Anstee theorem

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
to be the indegree-outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson [1] (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 [2] 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 [3] (1982) observed in a different context that it is suffient to have
. Berger [4] reinvented this result and gives a direct proof.

## Theorem statement

A sequence
of nonnegative integer pairs with
is digraphic if and only if
and the following inequality holds for *k* such that
:

## Stronger versions

Berger [4] proved that it suffices to consider the th inequality such that with and for .

## Other notations

The theorem can also be stated in terms of zero-one matrices. The connection can be seen if one realizes that each directed graph has an adjacency matrix where the column sums and row sums correspond to and . Note that the diagonal of the matrix only contains zeros. There is a connection to the relation majorization. We define a sequence with . Sequence can also be determined by a corrected Ferrers diagram. Consider sequences , and as -dimensional vectors , and . Since by applying the principle of double counting, the theorem above states that a pair of nonnegative integer sequences with nonincreasing is digraphic if and only if vector majorizes .

## Generalization

A sequence of nonnegative integer pairs with is digraphic if and only if and there exists a sequence such that the pair is digraphic and majorizes .[5]

## Characterizations for similar problems

Similar theorems describe the degree sequences of simple graphs, simple directed graphs with loops, and simple bipartite graphs. The first problem is characterized by the Erdős–Gallai theorem. The latter two cases, which are equivalent see Berger,[4] are characterized by the Gale–Ryser theorem.

## See also

## References

- D.R. Fulkerson:
*Zero-one matrices with zero trace.*In:*Pacific J. Math.*No. 12, 1960, pp. 831–836 - Wai-Kai Chen:
*On the realization of a (*p*,*s*)-digraph with prescribed degrees .*In:*Journal of the Franklin Institute*No. 6, 1966, pp. 406–422 - Richard Anstee:
*Properties of a class of (0,1)-matrices covering a given matrix.*In:*Can. J. Math.*, 1982, pp. 438–453 - Annabell Berger:
*A Note on the Characterization of Digraphic Sequences*In:*Discrete Mathematics*, 2014, pp. 38–41 - Annabell Berger:
*The connection between the number of realizations for degree sequences and majorization*In:*arXiv1212.5443*, 2012