## ★ Matching in hypergraphs

In graph theory, a hypergraph matching is a set of hypergraphs in which every two hypergraphs are disjoint. This is an extension of the concept of matching in the graph.

## 1. Definition. * (Определение) *

Recall that a hypergraph H is a pair of V, E, where V is the set of vertices, and E is the set of subsets of V, called hyperages. Each hyperedge can contain one or more vertexes.

A match in H is a subset M of E such that every two hypercurses e 1 and e 2 in M have an empty intersection and no common vertex.

The matching number of the hypergraph H is the largest matching size in H. it is often denoted V H {\displaystyle \nu H}.

As an example, let V be the set {1.2.3.4.5.6.7}. consider a 3-uniform hypergraph on a V hypergraph in which each hyperedge contains exactly 3 vertices. Let H-3 be a uniform hypergraph with 4 hyperreferences:

{ {1.2.3}, {1.4.5}, {4.5.6}, {2.3.6} }Then H admits several correspondences of size 2, for example:

{ {1.2.3}, {4.5.6} } { {1.4.5}, {2.3.6} }However, in any subset of 3 hyperfields, at least two of them intersect, so there is no correspondence of size 3. Hence, the corresponding number H is 2.

## 2. Correspondence in the graph as a special case. * (Соответствие в графе как частный случай) *

A graph without self-loops is just a 2-uniform hypergraph: each edge can be considered as a set of two vertices that it connects. For example, this 2-uniform hypergraph is a graph with 4 vertices {1.2.3.4} and 3 edges:

{ {1.3}, {1.4}, {2.4} }According to the above definition, a correspondence in a graph is a set of M edges, such that every two edges in M have an empty intersection. This is equivalent to saying that no two edges in M are adjacent to the same vertex, this is exactly the definition of matching in the graph.

## 3. Partial approval. * (Частичное одобрение) *

Fractional matching in a hypergraph is a function that assigns a fraction in each hyperedge, so that for each vertex v in V, the sum of the fractions of hyperedges containing v is no more than 1. a Match is a special case of fractional matching in which all fractions are equal to either 0 or 1. The size of a fractional match is the sum of fractions of all hyperedges.

The fractional correspondence number of a hypergraph H is the largest size of the fractional correspondence in H. it is often denoted by V * H {\displaystyle \nu ^{*}H}.

Since the mapping is a special case of fractional mapping, for each hypergraph H:

compliance is the number of H ≤ fractional matching-a number of H,

In characters:

ν H ≤ ν ∗ H. {\displaystyle \nu H\leq \nu ^{*}H.}

In General, a fractional matching number can be greater than the matching number. The Zoltan Furedi theorem provides upper bounds for the fractional match-numberH / match-numberH ratio:

- If each hyperedge in H contains at most r vertices, then V * H / V h ≤ r− 1 / r {\displaystyle \nu ^{*}H / \nu H\leq r-1 / r}. In particular, in a simple graph V * H / V H ≤ 3 / 2 {\displaystyle \nu ^{*}H / \nu H\leq 3 / 2}.
- Inequality is sharp: let H r-R be a uniform finite projective plane. Then V h r = 1 {\displaystyle \nu H_{r}=1}, since every two hyperreferences intersect, and V *H r = r− 1 / r {\displaystyle \nu ^{ * } H_{r}=r-1 / R} by a fractional correspondence that assigns a weight of 1 / r to each hyperemesis, this is a correspondence, since each vertex is contained in R hyperemeses, and its size is r -1 / r, since R 2 - r exist 1 hyperemia. Consequently, the ratio is exactly equal to r -1 / r.

- If H-R-partite vertices are divided into r parts and each hyperedge contains a vertex from each part, then V * H / V H ≤ r − 1. {\displaystyle \nu ^{*}H / \nu H\leq r-1.} In particular, in the bipartite graph V * H = V H {\displaystyle \nu ^{*}H=\nu H}.
- The inequality is sharp: let H r be a truncated projective plane of order r -1. Then V H r - = 1 {\displaystyle \nu H_{r -} =1} since every two hyperreferences intersect, and V * H r- = r-1 {\displaystyle \nu ^{*}H_{r -}=r-1} by a fractional correspondence that assigns a weight of 1 / r to each hyperreference, there are r 2 - r hyperreferences.
- If r is such that r is a uniform finite projective plane that does not exist, for example, r =7, then a stronger inequality holds: ν ∗ H / ν h ≤ r − 1 {\displaystyle \nu ^{*}H / \nu H\leq r-1}.

## 4. Perfect match. * (Идеальный матч) *

A match M is called perfect if each vertex v in V is contained in exactly one hyperedge M. This is a natural extension of the concept of perfect correspondence in the graph.

A fractional correspondence of M is called perfect if for each vertex v in V, the sum of the fractions of hyperedges in M containing v is exactly 1.

Consider a hypergraph H in which each hyperedge contains at most n vertices. If H admits a perfect fractional correspondence, then its fractional correspondence number is at least |V| / n. if each hyperedge in H contains exactly n vertices, then its fractional correspondence number is exactly |V|/ n.this is a generalization of the fact that in a graph the size of the ideal correspondence is |V|/2.

Given a set V of vertices, a set e of subsets V is called balanced if the hypergraph V, E admits a perfect fractional correspondence.

For example, if V = {1.2.3, a, b, c} and E = { {1, a}, {2, a}, {1, b}, {2, b}, {3, c}}, then E is balanced with a perfect fractional correspondence { 1 / 2, 1 / 2, 1 / 2, 1 / 2, 1 }.

There are various sufficient conditions for the existence of an ideal correspondence in a hypergraph:

- The ideal correspondence in hypergraphs of high degree is sufficient conditions analogous to Diracs theorem on Hamiltonian cycles based on the degree of vertices.
- Halls theorems for hypergraphs are sufficient conditions analogous to Halls marriage theorem based on sets of neighbors.

## 5. Intersecting hypergraph. * (Пересекающийся гиперграф) *

A hypergraph H = V, E is called intersecting if every two hyperrels in E have a common vertex. In the intersecting graph, there is no match with two or more hyperedges, so V H = 1 {\displaystyle \nu H=1}.

## 6. Balanced set-family. * (Сбалансированный набор-семья) *

A family of sets E over a basic set V is called balanced with respect to V If the hypergraph H = V, E admits a perfect fractional correspondence.

For example, consider the set of vertices V = {1.2.3, a, b, c} and the set of edges E = {1-a, 2-a, 1-b, 2-b, 3-c}. E is balanced because there is a perfect fractional correspondence with weights {1 / 2, 1 / 2, 1 / 2, 1 / 2, 1}.

## 7. The calculation for the maximum matching. * (Расчет для максимального соответствия) *

The problem of finding the correspondence of the maximum cardinality in a hypergraph, thus calculating V H {\displaystyle \nu H}, is NP-complete even for 3-homogeneous hypergraphs see 3-dimensional matting. This is in contrast to the case of simple 2-homogeneous graphs, in which the maximum power correspondence can be calculated in polynomial time.

## 8. Compliance and coverage. * (Соответствие требованиям и охват) *

A vertex cover in a hypergraph H = V, E is a subset T of V such that every hyperedge in E contains at least one vertex it is also called a transverse or impact set and is equivalent to a cover set. This is a generalization of the concept of vertex cover in a graph.

The vertex cover number of a hypergraph H is the smallest vertex cover size in H. it is often denoted τ H {\displaystyle \tau H}, for transverse.

A fractional vertex cover is a function that assigns a weight to each vertex in V such that for each hyperedge e in E, the sum of the vertex shares in e is at least 1. a Vertex cover is a special case of a fractional vertex cover in which all weights are either 0 or 1. The size of a fractional vertex cover is equal to the sum of fractions of all vertexes.

The fractional number of the vertex cover of a hypergraph H is the smallest size of the fractional vertex cover in H. it is often denoted τ ∗ H {\displaystyle \tau ^{*}H}.

Since vertex cover is a special case of fractional vertex cover, for every hypergraph H:

fractional vertex-cover number H ≤ vertex-cover number H.

The duality of linear programming implies that for every hypergraph H:

fractional matching number H = fractional vertex covering number H.

Hence, for every hypergraph H:

ν H ≤ ν ∗ H = τ ∗ H ≤ τ H {\displaystyle \nu H\leq \nu ^{*}H=\tau ^{*}H\leq \tau H}

If the size of each hyperge in H is at most r, then the Union of all hypergeases in maximum matching is a vertex cover. if there were an uncovered hyperge, we could add it to the match. Therefore:

τ H ≤ r ν H {\displaystyle \tau H\leq r\cdot \nu H}.

This inequality is a hard one: equality holds, for example, when V contains vertices r V H r-1 {\displaystyle r \ cdot \nu H r-1}, And E contains all subsets of vertices r.

However, in General case τ ∗ H <, r V H {\displaystyle \tau ^{*}H

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