IMDAD ULLAH KHAN UMM AL-QURA UNIVERSITY June 12, 2013€¦ · perfect matchings in uniform hypergraphs imdad ullah khan umm al-qura university june 12, 2013

PERFECT MATCHINGS IN UNIFORM HYPERGRAPHS

IMDAD ULLAH KHAN

UMM AL-QURA UNIVERSITY

June 12, 2013

Hypergraphs

A hypergraph H is a family of subsets (E (H)) of a ground setV (H)

H = (V ,E )

|V (H)| = n

H := E2 3

45

61

V = {1, 2, 3, 4, 5, 6}E = {{1, 5}, {1, 2, 3}, {2, 4, 5}, {1, 4, 5, 6}}

Hypergraphs: Terminology

A hypergraph is k-uniform if all edges are k-sets

H = (V ,E ), E ⊆(Vk

)

k-graphs

2-graphs are graphs

2 3

45

61

A k-graph is complete if all k-sets are edges

H = (V ,E ), E =(Vk

)

Hypergraphs: Terminology

H(V1,V2, . . . ,Vk) is a k-partite k-graph, if

V1,V2, . . . ,Vk is a partition of V (H)

Each edge uses one vertex from eachpart

V1

V2

V3

Complete k-partite k-graph

Balanced complete k-partite k-graph, Kr (t), t : size of eachpart.

Hypergraphs: Matching

A matching in a hypergraph is a set of disjoint edges

A perfect matching is a matching that covers all the vertices⌊n

k

⌋edges in k-graphs

n ∈ kZ

Hypergraphs: Degrees

H : k-graph, 1 ≤ d ≤ k − 1

S ∈(Vd

)

Degree of S is the number of edges containing S

dH(S) = |{e ∈ E : S ⊂ e}|minimum d-degree, δd(H) = min

S∈(Vd)

dH(S)

d = k − 1: δk−1(H)minimum co-degree

d = 1: δ1(H)minimum vertex degree

δ2(H) = 1

δ1(H) = 2

Degree Threshold for Perfect Matching

Sufficient conditions to ensure existence of perfect matching

Definition

md(k , n) = min{m : δd(H) ≥ m =⇒ H has a PM}

Theorem

m1(2, n) ≤ n2

n2 − 1

n2 + 1

Kn/2+1,n/2−1

Result is best possible: m1(2, n) = n2

.

Perfect Matching: codegree

even

A

B

|A| odd|A| ∼ n

2

δ3(H) ∼ n2 − k

Perfect Matching: codegree

Theorem

1 Kuhn-Osthus 2006

mk−1(k , n) ≤ n2

+ 3k2√

n log n

2 Rodl-Rucinski-Szemeredi 2006

mk−1(k , n) ≤ n2

+ C log n

3 Rodl-Rucinski-Schacht-Szemeredi2008

mk−1(k , n) ≤ n2

+ k/4

4 Rodl-Rucinski-Szemeredi 2009

mk−1(k , n) ≥ n2− k + {3

2, 2, 5

2, 3}

even

A

B

|A| odd|A| ∼ n

2

δ3(H) ∼ n2 − k

Perfect Matching: d-degree

Theorem (Pikhurko 2008)

For k2≤ d ≤ k − 1

md(k , n) ≤(

1

2+ ε

)(n − d

k − d

)

Theorem (Treglown-Zhao 2012)

For k2≤ d ≤ k − 1

md(k , n) ∼ 1

2

(n − d

k − d

)

Perfect Matching: vertex-degree

|A| = n3 − 1

δ1(H) =(n−12

)−(2n/32

)

A

B

Conjecture

1 ≤ d < k/2 md(k , n) ∼(

n − d

k − d

)−(

n − nk

+ 1− d

k − d

)

1 ≤ d < k/2 md(k , n) ∼(

1−(

k − 1

k

)k−d)(

n − d

k − d

)

Perfect Matching: Vertex Degree

Conjecture

1 ≤ d < k/2 md(k , n) ∼(

1−(

k − 1

k

)k−d)(

n − d

k − d

)

Theorem (Han-Person-Schacht 2009)

d <k

2md(k , n) ≤

(k − d

k+ ε

)(n − d

k − d

)

Theorem (Markstrom-Rucinski 2010)

d <k

2md(k , n) ≤

(k − d

k− 1

kk−1 + ε

)(n − d

k − d

)


Conjecture

1 ≤ d < k/2 md(k , n) ∼(

1−(

k − 1

k

)k−d)(

n − d

k − d

)

k = 3, d = 1 → 59. k = 4, d = 1 → 37

64. k = 5, d = 1 → 369

625.

Theorem (Han-Person-Schacht 2009)

m1(3, n) ≤(

5

9+ ε

)(n

2

)

Theorem (Markstrom- Rucinski 2010)

m1(4, n) ≤(

42

64+ ε

)(n

3

)


Theorem (K.)

If H is a 3-graph on n ≥ n0 vertices and

δ1(H) ≥(

n − 1

2

)−(

2n/3

2

)+ 1,

then H contains a perfect matching.

|A| = n3 − 1

δ1(H) =(n−12

)−(2n/32

)

A

B

Independently, Kuhn-Osthus-Treglown proved this.

In fact they proved a stronger result.


Theorem (Kuhn-Osthus-Treglown)

If H is a 3-graph on n ≥ n0 vertices, 1 ≤ m ≤ n/3, and

δ1(H) ≥(

n − 1

2

)−(

n −m

2

)+ 1,

then H contains a matching of size at least m.

|A| = m− 1

δ1(H) =(n−12

)−(n−m

2

)

A

B


Theorem (K.)


δ1(H) ≥(

n − 1

3

)−(

3n/4

3

)+ 1,


|A| = n4 − 1

δ1(H) =(n−13

)−(3n/43

)

A

B


Theorem (Alon-Frankl-Huang-Rodl-Rucinski-Sudakov 2012)

m1(4, n) ∼ 3764

(n−13

)

m2(5, n) ∼ 12

(n−23

)

m1(5, n) ∼ 369625

(n−14

)

m2(6, n) ∼ 6711296

(n−24

)

m3(7, n) ∼ 1n

(n−33

)


Theorem (K.)


δ1(H) ≥(

n − 1

2

)−(

2n/3

2

)+ 1,


|A| = n3 − 1

δ1(H) =(n−12

)−(2n/32

)

A

B

3-graphs - vertex degree: Proof Strategy

We consider two cases

1 H is close to the extremal construction

2 H is non-extremal

|A| ∼ n3 − 1 A

Bvery few suchedges are in H

3-graphs - vertex degree: Absorbing

Absorbing Technique

S ⊂ V , A matching M absorbs the set S if

∃ M ′ : V (M ′) = V (M) ∪ S .

S

s1

s2

s3

M

M ′

3-graphs - vertex degree: Absorbing Lemma

Absorbing Lemma (Han-Person-Schacht 2009)

If δ1(H) ≥(12

+ ε) (

nk

), then

∃ MA such that

|V (MA)| = ε1n and

∀ S : |S | = ε2n,MA is S-absorbing.

3-graphs - vertex degree: Proof Overview

Proof Outline:

1 Find a small absorbing matching MA (|V (MA)| ≤ ε1n)

2 Find an almost perfect matching M ′ in H − V (MA)

V0 = V (H)− (V (MA) + V (M ′)

|V0| ≤ ε2n

3 Absorb V0 into MA

V0

MA

M ′

3-graphs - vertex degree: Almost Perfect Matching:

V0

MA

M ′

We cover almost all graph with complete tripartite graphs

Theorem (Erdos 1964)

If |E (H)| ≥ ε(n3

), then H has K3(c

√log n).

Using this find as many K3(t)’s.

. . . Ivery few edges

Extend this to almost perfect cover.

3-graphs - vertex degree: Almost perfect cover



Suppose many pairs in I make edges with many verticesin two color classes of many tripartite graphs.

I I




The link graph:

I


Fact

Let B be a balanced bipartite graph on 6 vertices. If B has atleast 5 edges, then

B has a perfect matching or

B contains B320 as a subgraph or

B is isomorphic to B311.

B320PM B311



Few edges inside I and few pairs in I make edges withvertices in two color classes of many tripartite graphs.

δ1(H) implies that on average the link graph of a pair oftripartite graphs has 5 edges.

Suppose for many pairs the link graph has perfect matching.

ITi

Tj



Suppose for many pairs the link graph has a B320.

I

Tj

Ti



Few edges inside I.Few pairs in I make edges with vertices in two color classes ofmany tripartite graphs.

δ1(H) implies that on average the link graph of a pair oftripartite graphs has 5 edges.

For few pairs the link graph has perfect matching or has a B320.

V1

V2

V3

. . . . . .



For almost all pairs of tripartite graphs, the link graph isismorphic to B311.

V1

V2

V3

. . .I

Few edges in V2 ∪ V3

Similarly few edges with two vertices in I and one in V2 ∪ V3.


V1

V2

V3

. . .I

Few edges in IFew edges in V2 ∪ V3

Few edges with two vertices in I and one in V2 ∪ V3.

By definition of B311, few edges with one vertex in I andtwo in V2 ∪ V3.

So few edges in V2 ∪ V3 ∪ I, while its size is ∼ 2n/3, henceH is close to the extremal construction.


Thank You!

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IMDAD ULLAH KHAN UMM AL-QURA UNIVERSITY June 12, 2013€¦ · perfect matchings in uniform hypergraphs imdad ullah khan umm al-qura university june 12, 2013