Bisections and Edge-Disjoint 1-factors
Bisections and Edge-Disjoint 1-factors
Tyler Seacrest
Joint work with Stephen G. Hartke
October 11, 2010
Bisections and Edge-Disjoint 1-factors
Table of contents
1
Degree Sequences
Definition and Example
Potential k-factors
Kundu Generalization
2
Finding Large Bisections
Bollobás-Scott Conjecture
Potential Case
Application to Kundu Generalization
Approximate Version
Finding Edge-Disjoint 1-factors
3
Extending Bollobás-Scott
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Definition and Example
Degree Sequences
Let G be a graph. If you take the degree of each vertex
and put them into a sequence π, what you have is a
graphic degree sequence. We say that G realizes π.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Definition and Example
Degree Sequences
Let G be a graph. If you take the degree of each vertex
and put them into a sequence π, what you have is a
graphic degree sequence. We say that G realizes π.
Example
π = (4, 3, 2, 2, 2, 1),
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Definition and Example
Degree Sequences
Let G be a graph. If you take the degree of each vertex
and put them into a sequence π, what you have is a
graphic degree sequence. We say that G realizes π.
Example
π = (4, 3, 2, 2, 2, 1),
2
G
3
4
2
1
2
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Finding a 1-factor
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Finding a 1-factor
π = (4, 3, 2, 2, 2, 1)
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Finding a 1-factor
π = (4, 3, 2, 2, 2, 1)
2
G
3
4
2
1
2
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Finding a 1-factor
π = (4, 3, 2, 2, 2, 1)
2
G
3
4
2
1
2
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Potential k-factors
We say π potentially has a k-factor if some realization
of π has a k-factor.
Finding a 1-factor
π = (4, 3, 2, 2, 2, 1)
2
G
3
4
2
1
2
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Kundu’s k-factor Theorem
Theorem (Kundu 1973)
A graphic sequence π = (d1 , d2 , . . . , dn ) has a potential
k-factor if and only if π − k = (d1 − k, d2 − k, . . . , dn − k) is
also graphic.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Kundu’s k-factor Theorem
Theorem (Kundu 1973)
A graphic sequence π = (d1 , d2 , . . . , dn ) has a potential
k-factor if and only if π − k = (d1 − k, d2 − k, . . . , dn − k) is
also graphic.
π = (4, 3, 2, 2, 2, 1)
π − 1 = (3, 2, 1, 1, 1, 0)
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Kundu’s k-factor Theorem
Theorem (Kundu 1973)
A graphic sequence π = (d1 , d2 , . . . , dn ) has a potential
k-factor if and only if π − k = (d1 − k, d2 − k, . . . , dn − k) is
also graphic.
π − 1 = (3, 2, 1, 1, 1, 0)
π = (4, 3, 2, 2, 2, 1)
2
G
3
4
2
1
2
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Kundu’s k-factor Theorem
Theorem (Kundu 1973)
A graphic sequence π = (d1 , d2 , . . . , dn ) has a potential
k-factor if and only if π − k = (d1 − k, d2 − k, . . . , dn − k) is
also graphic.
π − 1 = (3, 2, 1, 1, 1, 0)
π = (4, 3, 2, 2, 2, 1)
2
G
3
4
2
1
2
1
H
2
3
1
0
1
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Potential k-factors
Kundu’s k-factor Theorem
Remark
There is no potential f -factor theorem.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Assume π has even length.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Assume π has even length.
Conjecture (Busch, Ferrara, Hartke, Jacobson, Kaul and
West)
A graphic degree sequence π potentially has k
edge-disjoint 1-factors if and only if π − k is graphic.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Assume π has even length.
Conjecture (Busch, Ferrara, Hartke, Jacobson, Kaul and
West)
A graphic degree sequence π potentially has k
edge-disjoint 1-factors if and only if π − k is graphic.
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Kundu Generalization
Theorem (Busch et. al.)
If graphic sequence π has minimum degree 2n + k − 2,
then π has a realization with k edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large regular bipartite subgraph.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large regular bipartite subgraph.
Split into 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large regular bipartite subgraph.
Split into 1-factors.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large balanced bipartite subgraph.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large balanced bipartite subgraph.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large balanced bipartite subgraph.
Find a large regular bipartite subgraph.
Bisections and Edge-Disjoint 1-factors
Degree Sequences
Kundu Generalization
Other Methods of Attack
Find a large balanced bipartite subgraph.
Find a large regular bipartite subgraph.
Split into 1-factors.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
Definition
A spanning balanced bipartite subgraph is called a
bisection.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
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
Ÿ
œ
degG (v)
.
degH (v) ≥
2
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Bollobás-Scott Conjecture
Finding Large Bisections
œ
degH (v) ≥
This is tight if true:
degG (v)
2
Ÿ
.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Large Potential Bisections
Conjecture (Bollobás-Scott 1999)
Every graph G has a bisection H such that for every
vertex v
œ
Ÿ
degG (v)
degH (v) ≥
.
2
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Large Potential Bisections
Conjecture (Bollobás-Scott 1999)
Every graph G has a bisection H such that for every
vertex v
œ
Ÿ
degG (v)
degH (v) ≥
.
2
Theorem (Potential Version, Hartke, S)
Every graphic sequence π has a realization G with a
bisection H such that for every vertex v
œ
Ÿ
degG (v) − 1
degH (v) ≥
.
2
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
We use a strengthened form of Havel-Hakimi due to
Kleitman and Wang.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
We use a strengthened form of Havel-Hakimi due to
Kleitman and Wang.
Theorem (Kleitman, Wang)
Fix any i. The sequence π is graphic if and only if, the
sequence
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
We use a strengthened form of Havel-Hakimi due to
Kleitman and Wang.
Theorem (Kleitman, Wang)
Fix any i. The sequence π is graphic if and only if, the
sequence
(d1 − 1, d2 − 1, . . . , ddi − 1,
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
We use a strengthened form of Havel-Hakimi due to
Kleitman and Wang.
Theorem (Kleitman, Wang)
Fix any i. The sequence π is graphic if and only if, the
sequence
(d1 − 1, d2 − 1, . . . , ddi − 1, ddi +1 , . . . , di+1 , di−1 ,
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott
Let π = (d1 ≥ d2 ≥ · · · ≥ dn ).
We use a strengthened form of Havel-Hakimi due to
Kleitman and Wang.
Theorem (Kleitman, Wang)
Fix any i. The sequence π is graphic if and only if, the
sequence
(d1 − 1, d2 − 1, . . . , ddi − 1, ddi +1 , . . . , di+1 , di−1 , . . . , dn )
is graphic.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott, Proof Outline
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott, Proof Outline
Every graphic sequence π has a realization G with a
bisection H such that for every vertex v
Ÿ
œ
degG (v) − 1
.
degH (v) ≥
2
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott, Proof Outline
Every graphic sequence π has a realization G with a
bisection H such that for every vertex v
Ÿ
œ
degG (v) − 1
.
degH (v) ≥
2
Let π = d1 ≥ · · · ≥ dn .
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott, Proof Outline
Every graphic sequence π has a realization G with a
bisection H such that for every vertex v
Ÿ
œ
degG (v) − 1
.
degH (v) ≥
2
Let π = d1 ≥ · · · ≥ dn .
Partition as so:
d1
d2
d3
d4
d5
d6
d6
d8
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
Potential Bollobás-Scott, Proof Outline
Every graphic sequence π has a realization G with a
bisection H such that for every vertex v
Ÿ
œ
degG (v) − 1
.
degH (v) ≥
2
Let π = d1 ≥ · · · ≥ dn .
Partition as so:
Use Kleitman-Wang to
add adjacencies.
d1
d2
d3
d4
d5
d6
d6
d8
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
A few more details
8
7
7
7
7
6
6
6
5
5
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
A few more details
7 8
7 6
6 7
7 6
6 7
6
6
6
5
5
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
A few more details
6 7 8
7 6
6 7
7 6
5 6 7
6 5
5 6
6 5
5
5
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Potential Case
A few more details
6 7 8
7 6
6 7
7 6
5 6 7
6 5
5 6
6 5
5
5
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
So we have a large potential
bisection.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
So we have a large potential
bisection.
Can we get a large regular
bisection from that?
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
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
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
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
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
Application to Kundu Generalization
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
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
So what do you get?
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Application to Kundu Generalization
So what do you get?
Theorem (Hartke, S)
Every graphic sequence of minimum degree δ at least
n/ 2 + 2 has a realization with
p
δ − 2 + n(2δ − n − 4)
2
edge-disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
We showed the following approximate version of
Bollobás-Scott (Independently, Albert Bush has a
similar result)
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
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) ≥
degG (v)
2
−
p
deg(v) ln n
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(n).
Proof Outline.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(n).
Proof Outline.
Arbitrarily pair the vertices.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(n).
Proof Outline.
Arbitrarily pair the vertices.
Randomly split each pair
between the two partite sets.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(n).
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.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(n).
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 union sum
bound .
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Approximate Version
Approximate Bollobás-Scott
degH (v) ≥
1
2
degG (v) −
p
degG (v) ln(3∆).
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.
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Finding Edge-Disjoint 1-factors
Edge-Disjoint 1-factors
Combining the bisection result with Csaba’s Theorem
on finding regular bipartite subgraphs, we get
Bisections and Edge-Disjoint 1-factors
Finding Large Bisections
Finding Edge-Disjoint 1-factors
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 +
least n/ 8 edge-disjoint 1-factors.
p
2n ln n has at
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Extending Bollobás-Scott
By extending the same probabilistic argument, we have
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Extending Bollobás-Scott
By extending the same probabilistic argument, we have
Theorem (Hartke, S)
Let G be a graph on n vertices, where n = pq for p > 1.
Then there exists a partition of the vertices of G into q
parts of size
pp such that every vertex v has at least
deg(v)/ q − deg(v) · ln(n) neighbors in each part.
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Example
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Example
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Example
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
Example
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
1-factors from Multipartite Theorem
Theorem (Harkte, S)
Any graph with n vertices, n and even square,
p of
minimum degree n/ 2 + (n ln n)3/ 4 has n/ 4 − n/ 4 edge
disjoint 1-factors.
Bisections and Edge-Disjoint 1-factors
Extending Bollobás-Scott
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