Bipartite hypergraph

In the weakest sense, a hypergraph H = (V, E) is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge meets both X and Y.

Equivalently, the vertices of H can be 2-colored so that no hyperedge is monochromatic. Therefore a hypergraph satisfying this condition is also called 2-colorable. A hypergraph in which some hyperedges are singletons (contain only one vertex) is obviously not 2-colorable; to avoid such trivial obstacles to 2-colorability, it is common to consider hypergraphs that are essentially 2-colorable, i.e., they become 2-colorable upon deleting all their singleton hyperedges.[1]^:468

The property fo 2-colorability was first introduced by Felix Bernstein in the context of set families;[2] therefore it is also called Property B.

In a stronger sense, a hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X.[3][4]

To see that this sense is stonger than 2-colorability, let H be a hypergraph on the vertices {1, 2, 3, 4} with the following hyperedges:

{ {1,2,3} , {1,2,4} , {1,3,4} , {2,3,4} }

This H is 2-colorable, for example by the partition X = {1,2} and Y = {3,4}. However, it is not exactly-2-colorable, since every set X with one element has an empty intersection with one hyperedge, and every set X with two or more elements has an intersection of size 2 or more with at least two hyperedges.

Hall's marriage theorem has been generalized from bipartite graphs to exactly-2-colorable hypergraphs; see Haxell's matching theorem.

Given an integer n, a hypergraph is called n-uniform if all its hyperedges contain exactly n vertices. An n-uniform hypergraph is called n-partite if its vertex set V can be partitioned into n subsets such that each hyperedge contains exactly one element from each subset. [5] An alternative term is rainbow-colorable.[6]

To see that n-partiteness is stronger than exact-2-colorability, let H be a hypergraph on the vertices {1, 2, 3, 4} with the following hyperedges;

{ {1,2,3} , {1,2,4} , {1,3,4} }

This H is 3-uniform. It is exactly-2-colorable by the partition X = {1} and Y = {2,3,4}. However, it is not 3-partite: in every partition of V into 3 subsets, at least one subset contains two vertices, and thus at least one hyperedge contains two vertices from this subset.

In another stronger sense, a hypergraph is called bipartite if it is essentially 2-colorable, and remains essentially 2-colorable upon deleting any number of vertices. A hypergraph satisfying this condition is also called balanced.[1]^:468

The see that this sense is stonger than 2-colorability, let H be the same hypergraph from the previous subsection:

{ {1,2,3} , {1,2,4} , {1,3,4} }

It is 2-colorable (it is even exactly-2-colorable), but it is not balanced. For example, if vertex 1 is removed, we get the restriction of H to {2,3,4}, which has the following hyperedges;

{ {2,3} , {2,4} , {3,4} }

It is not 2-colorable, since in any 2-coloring there are at least two vertices with the same color, and thus at least one of the hyperedges is monochromatic. The above example shows that exact 2-colorability does not imply balancedness; to see that balancedness does not imply exact-2-colorability either, let H be the hypergraph:[7]

{ {1,2} , {3,4} , {1,2,3,4} }

it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not exactly 2-colorable, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge.

Some theorems on bipartite graphs have been generalized to balanced hypergraphs.[8][9]^:468–470

• A hypergraph is balanced iff it does not contain an unbalanced odd-length circuit just like a simple graph is bipartite iff it does not contain an odd-length cycle).
• In every balanced hypergraph, the minimum vertex-cover has the same size as its maximum matching. This generalizes the Kőnig-Egervary theorem on bipartite graphs.
• In every balanced hypergraph, the degree (= the maximum number of hyperedges containing some one vertex) equals the chromatic index (= the least number of colors required for coloring the hyperedges such that no two hyperedges with the same color have a vertex in common).[10] This generalizes a theorem of Konig on bipartite graphs.

A graph is a special case of a hypergraph in which all hyperedges contain 2 vertices.

In this special case, all the above definitions are equivalent: a graph is 2-colorable iff it is exactly-bipartite iff it is rainbow-colorable iff it is balanced iff it is a bipartite graph.

• Matching in hypergraphs
• Vertex cover in hypergraphs