IMDAD ULLAH KHAN UMM AL-QURA UNIVERSITY June 12, 2013 Hypergraphs - - PowerPoint PPT Presentation

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 set V (H) H = (V , E) |V (H)| = n H := E

2 3 4 5 6 1

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 ⊆ V

k

• k-graphs

2-graphs are graphs

2 3 4 5 6 1

A k-graph is complete if all k-sets are edges H = (V , E), E = V

k

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 each part

V1 V2 V3

Complete k-partite k-graph Balanced complete k-partite k-graph, Kr(t), t : size of each part.

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 ∈ V

d

• Degree of S is the number of edges containing S

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

S∈(V

d)

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) ≤ n

2

n 2 − 1 n 2 + 1

Kn/2+1,n/2−1

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

2.

Perfect Matching: codegree

even A B |A| odd |A| ∼ n

2

δ3(H) ∼ n

2 − k

Perfect Matching: codegree

Theorem

1 K¨

uhn-Osthus 2006 mk−1(k, n) ≤ n

2 + 3k2√n log n 2 R¨

• dl-Ruci´

nski-Szemer´ edi 2006 mk−1(k, n) ≤ n

2 + C log n 3 R¨

• dl-Ruci´

nski-Schacht-Szemer´ edi 2008 mk−1(k, n) ≤ n

2 + k/4 4 R¨

• dl-Ruci´

nski-Szemer´ edi 2009 mk−1(k, n) ≥ n

2 − k + { 3 2, 2, 5 2, 3} even A B |A| odd |A| ∼ n

2

δ3(H) ∼ n

2 − k

Perfect Matching: d-degree

Theorem (Pikhurko 2008) For k

2 ≤ d ≤ k − 1

md(k, n) ≤ 1 2 + ǫ n − d k − d

• Theorem (Treglown-Zhao 2012)

For k

2 ≤ d ≤ k − 1

md(k, n) ∼ 1 2 n − d k − d

Perfect Matching: vertex-degree

|A| = n

3 − 1

δ1(H) = n−1

2

• −

2n/3

2

• A

B

Conjecture 1 ≤ d < k/2 md(k, n) ∼ n − d k − d

• −

n − n

k + 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 (H`

an-Person-Schacht 2009) d < k 2 md(k, n) ≤ k − d k + ǫ n − d k − d

• Theorem (Markstr¨
• m-Ruci´

nski 2010) d < k 2 md(k, n) ≤ k − d k − 1 kk−1 + ǫ 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

• k = 3, d = 1 → 5
• 9. k = 4, d = 1 → 37
• 64. k = 5, d = 1 → 369

625.

Theorem (H` an-Person-Schacht 2009) m1(3, n) ≤ 5 9 + ǫ n 2

• Theorem (Markstr¨
• m- Ruci´

nski 2010) m1(4, n) ≤ 42 64 + ǫ n 3

Perfect Matching: Vertex Degree

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| = n

3 − 1

δ1(H) = n−1

2

• −

2n/3

2

• A

B

Independently, K¨ uhn-Osthus-Treglown proved this. In fact they proved a stronger result.

Perfect Matching: Vertex Degree

Theorem (K¨ uhn-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−1

2

• −

n−m

2

• A

B

Perfect Matching: Vertex Degree

Theorem (K.) If H is a 4-graph on n ≥ n0 vertices and δ1(H) ≥ n − 1 3

• −

3n/4 3

• + 1,

then H contains a perfect matching. |A| = n

4 − 1

δ1(H) = n−1

3

• −

3n/4

3

• A

B

Perfect Matching: Vertex Degree

Theorem (Alon-Frankl-Huang-R¨

• dl-Ruci´

nski-Sudakov 2012) m1(4, n) ∼ 37

64

n−1

3

• m2(5, n) ∼ 1

2

n−2

3

• m1(5, n) ∼ 369

625

n−1

4

• m2(6, n) ∼

671 1296

n−2

4

• m3(7, n) ∼ 1

n

n−3

3

Perfect Matching: Vertex Degree

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| = n

3 − 1

δ1(H) = n−1

2

• −

2n/3

2

• 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| ∼ n

3 − 1

A B very few such edges 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 (H` an-Person-Schacht 2009) If δ1(H) ≥ 1

2 + ǫ

n

k

• , then

∃ MA such that |V (MA)| = ǫ1n and ∀ S : |S| = ǫ2n, MA is S-absorbing.

3-graphs - vertex degree: Proof Overview

Proof Outline:

1 Find a small absorbing matching MA (|V (MA)| ≤ ǫ1n) 2 Find an almost perfect matching M′ in H − V (MA)

V0 = V (H) − (V (MA) + V (M′) |V0| ≤ ǫ2n

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 (Erd˝

• s 1964)

If |E(H)| ≥ ǫ n

3

• , then H has K3(c√log n).

Using this find as many K3(t)’s.

. . . I

very few edges Extend this to almost perfect cover.

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

Extend this to almost perfect cover. Suppose many pairs in I make edges with many vertices in two color classes of many tripartite graphs.

I I

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

Extend this to almost perfect cover. The link graph:

I

3-graphs - vertex degree: Almost perfect cover

Fact Let B be a balanced bipartite graph on 6 vertices. If B has at least 5 edges, then B has a perfect matching or B contains B320 as a subgraph or B is isomorphic to B311.

B320 PM B311

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

Few edges inside I and few pairs in I make edges with vertices in two color classes of many tripartite graphs. δ1(H) implies that on average the link graph of a pair of tripartite graphs has 5 edges. Suppose for many pairs the link graph has perfect matching.

I

Ti Tj

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

Suppose for many pairs the link graph has a B320.

I

Tj Ti

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

Few edges inside I. Few pairs in I make edges with vertices in two color classes of many tripartite graphs. δ1(H) implies that on average the link graph of a pair of tripartite graphs has 5 edges. For few pairs the link graph has perfect matching or has a B320.

V1 V2 V3

. . . . . .

3-graphs - vertex degree: Almost perfect cover

. . . I

very few edges

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

V1 V2 V3

. . .

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

3-graphs - vertex degree: Almost perfect cover

V1 V2 V3

. . .

I

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

3-graphs - vertex degree: Almost perfect cover

Thank You!