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 sufficient to have

Berger [4] reinvented this result and gives a direct proof.

A sequence

of nonnegative integer pairs with

is digraphic if and only if

and the following inequality holds for k such that

: Berger proved[4] that it suffices to consider the

th inequality such that

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

Note that the diagonal of the matrix only contains zeros.

There is a connection to the relation majorization.

We define a sequence

can also be determined by a corrected Ferrers diagram.

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

of nonnegative integer pairs with

is digraphic if and only if

and there exists a sequence

such that the pair

is digraphic and

[5] 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.