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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
)
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 → 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
)
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| = n3 − 1
δ1(H) =(n−12
)−(2n/32
)
A
B
Independently, Kuhn-Osthus-Treglown proved this.
In fact they proved a stronger result.
Perfect Matching: Vertex Degree
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
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| = n4 − 1
δ1(H) =(n−13
)−(3n/43
)
A
B
Perfect Matching: Vertex Degree
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
)
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| = 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
. . . Ivery few edges
Extend this to almost perfect cover.
Suppose many pairs in I make edges with many verticesin two color classes of many tripartite graphs.
I I
3-graphs - vertex degree: Almost perfect cover
. . . Ivery 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 atleast 5 edges, then
B has a perfect matching or
B contains B320 as a subgraph or
B is isomorphic to B311.
B320PM B311
3-graphs - vertex degree: Almost perfect cover
. . . Ivery few edges
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
3-graphs - vertex degree: Almost perfect cover
. . . Ivery few edges
Suppose for many pairs the link graph has a B320.
I
Tj
Ti
3-graphs - vertex degree: Almost perfect cover
. . . Ivery few edges
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
. . . . . .
3-graphs - vertex degree: Almost perfect cover
. . . Ivery few edges
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.
3-graphs - vertex degree: Almost perfect cover
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.
3-graphs - vertex degree: Almost perfect cover
Thank You!