In graph theory, perfect matching in high-degree hypergraphs is a research avenue trying to find sufficient conditions for existence of a perfect matching in a hypergraph, based only on the degree of vertices or subsets of them.
In a simple graph, the degree of a vertex, often denoted by or, is the number of edges in adjacent to . The minimum degree of a graph, often denoted by or, is the minimum of over all vertices in .
A matching in a graph is a set of edges such that each vertex is adjacent to at most one edge; a perfect matching is a matching in which each vertex is adjacent to exactly one edge. A perfect matching does not always exist, and thus it is interesting to find sufficient conditions that guarantee its existence.
One such condition follows from Dirac's theorem on Hamiltonian cycles. It says that, if, then the graph admits a Hamiltonian cycle; this implies that it admits a perfect matching. The factor is tight, since the complete bipartite graph on vertices has degree but does not admit a perfect matching.
The results described below aim to extend these results from graphs to hypergraphs.
In a hypergraph, each edge of may contain more than two vertices of . The degree of a vertex in is, as before, the number of edges in that contain . But in a hypergraph we can also consider the degree of subsets of vertices: given a subset of, is the number of edges in that contain all vertices of . Thus, the degree of a hypergraph can be defined in different ways depending on the size of subsets whose degree is considered.
Formally, for every integer, is the minimum of over all subsets of that contain exactly vertices. Thus, corresponds to the definition of a degree of a simple graph, namely the smallest degree of a single vertex; is the smallest degree of a pair of vertices; etc.
A hypergraph is called -uniform if every hyperedge in contains exactly vertices of . In -uniform graphs, the relevant values of are . In a simple graph, .
Several authors proved sufficient conditions for the case, i.e., conditions on the smallest degree of a single vertex.
\deg1(H)\geq(5/9+o(1)){n\choose2},
then contains a perfect matching.[1]
\deg1(H)\geq{n-1\choose2}-{2n/3\choose2}+1,
then contains a perfect matching - a matching of size . This result is the best possible.[2]
\deg1(H)\geq{n-1\choose3}-{3n/4\choose3},
then contains a perfect matching - a matching of size . This result is the best possible.[3]
\deg1(H)>
| r-1 | |
| r |
\left({\binom{n-1}{r-1}-\binom{n-kr-1}{r-1}}\right),
then contains a matching of size at least . In particular, setting gives that, if
\deg1(H)>
| r-1 | |
| r |
\left({\binom{n-1}{r-1}-1}\right),
then contains a perfect matching.[4]
\deg1(H)>
| r-1 | |
| r |
\left({nr-1-(n-k)r-1
then contains a matching of size at least . In particular, setting gives that if
\deg1(H)>
| r-1 | |
| r |
\left({nr-1-1}\right),
then contains a perfect matching.
For comparison, Dirac's theorem on Hamiltonian cycles says that, if is 2-uniform (i.e., a simple graph) and
\deg1(H)\geq{n-1\choose1}-{n/2\choose1}+1=n/2,
Several authors proved sufficient conditions for the case, i.e., conditions on the smallest degree of sets of vertices, in -uniform hypergraphs with vertices.
The following results relate to -partite hypergraphs that have exactly vertices on each side (vertices overall):
\degr-1(H)\geqn/2+\sqrt{2nlog2n}
and, then has a perfect matching. This expression is smallest possible up to the lower-order term; in particular, is not sufficient.
\degr-1(H)\geqn/r
then admits a matching that covers all but at most vertices in each vertex class of . The factor is essentially the best possible.
\degr-1(H)\geq(1/2+\gamma)n
then is Hamiltonian, and thus contains a perfect matching.[6]
\degr-1(H)\geqn/2+3r2\sqrt{nlog2n}
and is sufficiently large, then admits a perfect matching.
\degr-1(H)\geqn/r+3r2\sqrt{nlog2n},
then admits a matching that covers all but at most vertices. [7]
\degr-1(H)\geqn/2-r+ck,n
where is a constant depending on the parity of and (all expressions below are the best possible):[8]
There are some sufficient conditions for other values of :
| \left( | 1 |
| 2 |
+o(1)\right)\binom{n}{r-d}.
| \left( | r-d |
| r |
+o(1)\right)\binom{n-d}{r-d}
\degd(H)\geq
| r-d | |
| r |
nr-d+rnr-d-1,
then contains a matching covering all but vertices.
\degd(H)\geq
| r-d | |
| r |
\binom{n}{r-d}+rr+1(ln{n})1/2nr-d-1/2,
then contains a matching covering all but vertices.