The Gale–Ryser theorem is a result in graph theory and combinatorial matrix theory, two branches of combinatorics.
It provides one of two known approaches to solving the bipartite realization problem, i.e. it gives a necessary and sufficient condition for two finite sequences of natural numbers to be the degree sequence of a labeled simple bipartite graph; a sequence obeying these conditions is called "bigraphic".
It is an analog of the Erdős–Gallai theorem for simple graphs.
The theorem was published independently in 1957 by H. J. Ryser and David Gale.
A pair of sequences of nonnegative integers
: Sometimes this theorem is stated with the additional constraint
This condition is not necessary, because the labels of vertices of one partite set in a bipartite graph can be rearranged arbitrarily.
In 1962 Ford and Fulkerson [1] gave a different but equivalent formulation of the theorem.
The theorem can also be stated in terms of zero-one matrices.
The connection can be seen if one realizes that each bipartite graph has a biadjacency matrix where the column sums and row sums correspond to
Each sequence can also be considered as an integer partition of the same number
The conjugate partition can be determined by a Ferrers diagram.
, the theorem above states that a pair of nonnegative integer sequences a and b with nonincreasing a is bigraphic if and only if the conjugate partition
A third formulation is in terms of degree sequences of simple directed graphs with at most one loop per vertex.
the indegree-outdegree pairs of a labeled directed graph with at most one loop per vertex?
The theorem can easily be adapted to this formulation, because there does not exist a special order of b.
The proof is composed of two parts: the necessity of the condition and its sufficiency.
We outline the proof of both parts in the language of matrices.
To see that the condition in the theorem is necessary, consider the adjacency matrix of a bigraphic realization with row sums
, and shift all ones in the matrix to the left.
The operation of shifting all ones to the left increases a partition in majorization order, and so
The original proof of sufficiency of the condition was rather complicated.
Krause (1996) gave a simple algorithmic proof.
The idea is to start with the Ferrers diagram of
and shift ones to the right until the column sums are
steps, in each of which a single one entry is moved to the right.
Berger proved[2] that it suffices to consider those
A pair of finite sequences of nonnegative integers
This yields to the result that regular sequences have for fixed numbers of vertices and edges the largest number of bigraphic realizations, if n divides m. They are the contrary sequences of threshold sequences with only one unique bigraphic realization, which is known as threshold graph.
Minconvex sequences generalize this concept if n does not divide m. Similar theorems describe the degree sequences of simple graphs and simple directed graphs.