Large 1-factorizable subgraphs
Tyler Seacrest
Joint work with Stephen G. Hartke
University of Nebraska–Lincoln
January 8, 2011
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Edge-disjoint 1-factors
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Table of contents
1
Introduction
2
Finding Large Bisections
Bollobás-Scott Conjecture
Approximate Version
Finding Edge-Disjoint 1-factors
3
Extending Bollobás-Scott
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Edge-disjoint 1-factors
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Matching Students
Suppose you are teaching class (even number of students),
and want students to work in pairs, repeatedly. You want:
No two students work together twice.
Every student works with every other students.
Can you do it?
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Matching Students
Solution:
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Edge-disjoint 1-factors
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Some Definitions
Let G be a simple graph on n vertices where n is even.
A matching is a collection of disjoint edges of G.
The matching is perfect if every vertex is in one of the
edges of the matching.
A spanning regular subgraph, where every vertex has
degree k, is a k-factor. For example, a 1-factor is the same
thing as a perfect matching.
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Edge-disjoint 1-factors
January 8, 2011
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Possible Generalizations
We have shown that the complete graph G = Kn can be
decomposed into n − 1 edge-disjoint perfect matchings.
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Edge-disjoint 1-factors
January 8, 2011
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Possible Generalizations
We have shown that the complete graph G = Kn can be
decomposed into n − 1 edge-disjoint perfect matchings.
Question
What other regular graphs, where each vertex has degree d,
can be decomposed into d edge-disjoint perfect matchings?
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
6 / 22
Possible Generalizations
We have shown that the complete graph G = Kn can be
decomposed into n − 1 edge-disjoint perfect matchings.
Question
What other regular graphs, where each vertex has degree d,
can be decomposed into d edge-disjoint perfect matchings?
This is equivalent to determining the edge chromatic number
of G, which is d if and if it can be decomposed into
edge-disjoint perfect matchings.
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
6 / 22
Possible Generalizations
We have shown that the complete graph G = Kn can be
decomposed into n − 1 edge-disjoint perfect matchings.
Question
What other regular graphs, where each vertex has degree d,
can be decomposed into d edge-disjoint perfect matchings?
This is equivalent to determining the edge chromatic number
of G, which is d if and if it can be decomposed into
edge-disjoint perfect matchings.
Conjecture (1-factorization conjecture)
Every regular graph of degree n/ 2 or greater can be
decomposed into perfect matchings.
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Edge-disjoint 1-factors
January 8, 2011
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Edge-Disjoint Matchings
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
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Edge-disjoint 1-factors
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Edge-Disjoint Matchings
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
Why minimum degree n/ 2?
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Edge-disjoint 1-factors
January 8, 2011
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Edge-Disjoint Matchings
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
Why minimum degree n/ 2? The slightly unbalanced complete
bipartite graph has minimum degree n/ 2 − 1, and yet has no
perfect matchings at all.
.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
This is related to finding the largest k-factor in G.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
This is related to finding the largest k-factor in G.
Remark
It is easier to find a k-factor than it is to find k edge-disjoint
1-factors.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
This is related to finding the largest k-factor in G.
Remark
It is easier to find a k-factor than it is to find k edge-disjoint
1-factors.
This is also related to the question of finding edge-disjoint
Hamiltonian cycles.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Question
Given a graph G of minimum degree at least n/ 2, how many
edge-disjoint perfect matchings does it contain?
This is related to finding the largest k-factor in G.
Remark
It is easier to find a k-factor than it is to find k edge-disjoint
1-factors.
This is also related to the question of finding edge-disjoint
Hamiltonian cycles.
Remark
It is easier to find k edge-disjoint 1 factors than it is to find k/ 2
edge-disjoint Hamiltonian cycles.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
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Edge-disjoint 1-factors
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k-factors and Hamiltonian Cycles
Let G be a graph on n vertices, n even, with minimum degree
δ.
Theorem (Katerinis ‘85, Egawa and Enomoto ‘89)
If δ ≥ n/ 2, then G contains a k-factor for all k ≤ (n + 5)/ 4. This
is best possible.
Thus, roughly, n/ 4 edge-disjoint 1-factors is the best we can
hope for.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Theorem (Nash-Williams, ‘71)
If δ ≥ n/ 2, then G contains at least b5n/ 224c edge-disjoint
Hamiltonian cycles.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Theorem (Nash-Williams, ‘71)
If δ ≥ n/ 2, then G contains at least b5n/ 224c edge-disjoint
Hamiltonian cycles.
This gives roughly n/ 46 edge-disjoint Hamiltonian cycles,
which gives n/ 23 edge-disjoint 1-factors.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Theorem (Christofides, Kühn, Osthus, (to appear))
For every ε > 0, if n is sufficiently large and δ ≥ (1/ 2 + ε)n,
then G contains n/ 8 Hamiltonian cycles.
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Edge-disjoint 1-factors
January 8, 2011
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k-factors and Hamiltonian Cycles
Theorem (Christofides, Kühn, Osthus, (to appear))
For every ε > 0, if n is sufficiently large and δ ≥ (1/ 2 + ε)n,
then G contains n/ 8 Hamiltonian cycles.
Thus, given the restrictions of this theorem, we can achieve
n/ 4 edge-disjoint 1-factors. This theorem uses the regularity
lemma.
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Edge-disjoint 1-factors
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Other Methods of Attack
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Edge-disjoint 1-factors
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Other Methods of Attack
Find a large balanced bipartite subgraph.
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Edge-disjoint 1-factors
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Other Methods of Attack
Find a large balanced bipartite subgraph.
Find a large regular bipartite subgraph.
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Edge-disjoint 1-factors
January 8, 2011
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Other Methods of Attack
Find a large balanced bipartite subgraph.
Find a large regular bipartite subgraph.
Split into 1-factors.
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Edge-disjoint 1-factors
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Finding Large Bisections
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Edge-disjoint 1-factors
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Finding Large Bisections
Definition
A spanning balanced bipartite subgraph is called a bisection.
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Edge-disjoint 1-factors
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Finding Large Bisections
Definition
A spanning balanced bipartite subgraph is called a bisection.
Conjecture (Bollobás-Scott 2002)
Every graph G has a bisection H such that for every vertex v
œ
Ÿ
degG (v)
degH (v) ≥
.
2
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Edge-disjoint 1-factors
January 8, 2011
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Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
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Edge-disjoint 1-factors
January 8, 2011
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Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
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Edge-disjoint 1-factors
January 8, 2011
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Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
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Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
14 / 22
Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
14 / 22
Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
14 / 22
Finding Large Bisections
œ
degH (v) ≥
degG (v)
Ÿ
.
2
This is tight if true:
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
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Approximate Bollobás-Scott
We showed the following approximate version of
Bollobás-Scott (Independently, Albert Bush has a similar
result)
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Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
We showed the following approximate version of
Bollobás-Scott (Independently, Albert Bush has a similar
result)
Theorem
Every graph G has a bisection H where every vertex v satisfies
degH (v) ≥
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degG (v)
2
−
p
Edge-disjoint 1-factors
deg(v) ln n
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Approximate Bollobás-Scott
degH (v) ≥
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1
degG (v) −
p
2
Proof Outline.
degG (v) ln(n).
Edge-disjoint 1-factors
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Approximate Bollobás-Scott
degH (v) ≥
1
degG (v) −
p
2
Proof Outline.
degG (v) ln(n).
Arbitrarily pair the vertices.
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Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
degH (v) ≥
1
degG (v) −
p
2
Proof Outline.
degG (v) ln(n).
Arbitrarily pair the vertices.
Randomly split each pair
between the two partite sets.
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Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
degH (v) ≥
1
degG (v) −
p
2
Proof Outline.
degG (v) ln(n).
Arbitrarily pair the vertices.
Randomly split each pair
between the two partite sets.
Bound the probability of a bad
vertex using Chernoff bounds.
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Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
degH (v) ≥
1
degG (v) −
p
2
Proof Outline.
degG (v) ln(n).
Arbitrarily pair the vertices.
Randomly split each pair
between the two partite sets.
Bound the probability of a bad
vertex using Chernoff bounds.
Combine using the union sum
bound .
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Edge-disjoint 1-factors
January 8, 2011
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Approximate Bollobás-Scott
degH (v) ≥
1
degG (v) −
p
degG (v) ln(3∆).
2
Proof Outline.
Arbitrarily pair the vertices.
Randomly split each pair
between the two partite sets.
Bound the probability of a bad
vertex using Chernoff bounds.
Combine using the Lovász
Local Lemma.
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Edge-disjoint 1-factors
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Finding Edge-Disjoint 1-factors
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Edge-disjoint 1-factors
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Finding Edge-Disjoint 1-factors
So we have a large potential
bisection.
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Edge-disjoint 1-factors
January 8, 2011
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Finding Edge-Disjoint 1-factors
So we have a large potential
bisection.
Can we get a large regular
bisection from that?
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Edge-disjoint 1-factors
January 8, 2011
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Finding Edge-Disjoint 1-factors
So we have a large potential
bisection.
Can we get a large regular
bisection from that?
Theorem (Csaba 2007)
Every balanced bipartite graph, each part size n, of minimum
degree δ, with δ ≥ n/ 2, has a regular spanning subgraph of
size at least
p
δ + 2δn − n2
2
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Edge-disjoint 1-factors
January 8, 2011
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Finding Edge-Disjoint 1-factors
So we have a large potential
bisection.
Can we get a large regular
bisection from that?
Theorem (Csaba 2007)
Every balanced bipartite graph, each part size n, of minimum
degree δ, with δ ≥ n/ 2, has a regular spanning subgraph of
size at least
p
δ + 2δn − n2
2
T. Seacrest (UNL)
Edge-disjoint 1-factors
January 8, 2011
17 / 22
Finding Edge-Disjoint 1-factors
So we have a large potential
bisection.
Can we get a large regular
bisection from that?
Theorem (Csaba 2007)
Every balanced bipartite graph, each part size n, of minimum
degree δ, with δ ≥ n/ 2, has a regular spanning subgraph of
size at least
p
δ + 2δn − n2
2
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Edge-disjoint 1-factors
January 8, 2011
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Edge-Disjoint 1-factors
Combining the bisection result with Csaba’s Theorem on
finding regular bipartite subgraphs, we get
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Edge-disjoint 1-factors
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Edge-Disjoint 1-factors
Combining the bisection result with Csaba’s Theorem on
finding regular bipartite subgraphs, we get
Theorem (Hartke, S)
Any graph of minimum degree n/ 2 +
edge-disjoint 1-factors.
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p
Edge-disjoint 1-factors
2n ln n has at least n/ 8
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Extending Bollobás-Scott
Here, we split the graph into two parts such that every vertex
had many edges in each part. But there is nothing special
about two.
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Edge-disjoint 1-factors
January 8, 2011
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Extending Bollobás-Scott
Here, we split the graph into two parts such that every vertex
had many edges in each part. But there is nothing special
about two.
Theorem (Hartke, S)
Let G be a graph on n vertices. Then there exists a partition of
the vertices of G into q parts of size
p p or p + 1 such that every
vertex v has at least deg(v)/ q − deg(v) · ln(n) neighbors in
each part.
We can use this to get even more edge-disjoint perfect
matchings.
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Example
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Example
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Edge-disjoint 1-factors
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Example
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Edge-disjoint 1-factors
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Example
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1-factors from Multipartite Theorem
Theorem (Harkte, S)
Any graph with n vertices, of minimum degree
n/ 2 + O[(n ln n)3/ 4 ] has n/ 4 edge disjoint 1-factors.
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Edge-disjoint 1-factors
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Thanks!
Thanks!
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