Matching in hypergraphs
set of hyperedges where every pair is disjoint
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
Definition
Recall that a hypergraph H is a pair (V, E), where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices.
A matching in H is a subset M of E, such that every two hyperedges e1 and e2 in M have an empty intersection (have no vertex in common).
The matching number of a hypergraph H is the largest size of a matching in H. It is often denoted by ν(H).
As an example, let V be the set {1,2,3,4,5,6,7}. Consider a 3-uniform hypergraph on V (a hypergraph in which each hyperedge contains exactly 3 vertices). Let H be a 3-uniform hypergraph with 4 hyperedges:
{ {1,2,3}, {1,4,5}, {4,5,6}, {2,3,6} }
Then H admits several matchings of size 2, for example:
{ {1,2,3}, {4,5,6} }
{ {1,4,5}, {2,3,6} }
However, in any subset of 3 hyperedges, at least two of them intersect, so there is no matching of size 3. Hence, the matching number of H is 2.
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