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Concept
Bipartite hypergraph
Résumé
In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph.
Property B
The weakest definition of bipartiteness is also called 2-colorability. A hypergraph H = (V, E) is called 2-colorable 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. Every bipartite graph G = (X+Y, E) is 2-colorable: each edge contains exactly one vertex of X and one vertex of Y, so e.g. X can be colored blue and Y can be colored yellow and no edge is monochromatic.
The property of 2-colorability was first introduced by Felix Bernstein in the context of set families; therefore it is also called Property B.
A stronger definition of bipartiteness is: 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. Every bipartite graph is also a bipartite hypergraph.
Every bipartite hypergraph is 2-colorable, but bipartiteness is stronger 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 bipartite, 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 bipartite hypergraphs; see Hall-type theorems for hypergraphs.
An stronger definition is: 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.
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