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Combinatorics Using Computational Methods
Derrick Stolee
University of Nebraska–[email protected]
http://www.math.unl.edu/∼s-dstolee1/
March 13, 2012Dissertation Defense
Supported by NSF grants DMS-0354008, DMS-0914815, and CCF-0916525,
and a University of Nebraska Presidential Fellowship.
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Advisors and Committee
Stephen Hartke Vinod VariyamMathematics Computer Science
and Engineering
Jamie Radcliffe Stephen Scott Christina FalciMathematics CSE Sociology
Derrick Stolee (UNL) Computational Combinatorics 2 / 68
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Thanks to...
Katie StoleeEric Allender, Pranav Anand, David Barrington, Brian Bockleman,Chris Bourke, Jane Butterfield, James Carraher, Henry Escuadro,Michael Ferrara, Lance Fortnow, Brady Garvin, Ralucca Gera, JoeGeisbauer, Ellen Gethner, Steve Goddard, Adam S. Jobson, TravisJohnston, Andre Kezdy, Elizabeth Kupin, Timothy D. LeSaulnier, JaredNishikawa, Kevin G. Milans, Andrew Ray, Ben Reiniger, Tyler Seacrest,Hannah (Kolb) Spinoza, Brendon Stanton, David Swanson, RaghunathTewari, Judy Walker, Derek Weitzel, Paul S. Wenger, Douglas B. West,Zahava Wilstein, Matthew Yancey, and
all UNL Math or CSE graduate students, staff, and faculty.
Full Acknowledgements available on my web page.
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Derrick Stolee (UNL) Computational Combinatorics 6 / 68
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
Structure
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
Structure
Perspective
Derrick Stolee (UNL) Computational Combinatorics 6 / 68
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
Structure
Perspective
Efficiency
Derrick Stolee (UNL) Computational Combinatorics 6 / 68
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
Structure
Perspective
Efficiency
Theorems
Derrick Stolee (UNL) Computational Combinatorics 6 / 68
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Overview The Big Question
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Problem
Examples
Structure
Perspective
Efficiency
Theorems
Computational Combinatorics
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Overview The Big Question
The Goal
Determine if certain combinatorial objects exist with givenstructural or extremal properties.
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Overview The Big Question
The Goal
Determine if certain combinatorial objects exist with givenstructural or extremal properties.
Examples:1 Is there a projective plane of order 10?
(Lam, Thiel, Swiercz, 1989)
2 When do strongly regular graphs exist?(Spence 2000, Coolsaet, Degraer, Spence 2006, many others)
3 How many Steiner triple systems are there of order 19?(Kaski, Ostergard, 2004)
Derrick Stolee (UNL) Computational Combinatorics 7 / 68
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Overview The Big Question
The Goal
Determine if certain combinatorial objects exist with givenstructural or extremal properties.
Examples:1 Is there a projective plane of order 10?
(Lam, Thiel, Swiercz, 1989)
2 When do strongly regular graphs exist?(Spence 2000, Coolsaet, Degraer, Spence 2006, many others)
3 How many Steiner triple systems are there of order 19?(Kaski, Ostergard, 2004)
Derrick Stolee (UNL) Computational Combinatorics 7 / 68
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Overview The Big Question
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger)
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy)
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey)
4 What graphs are uniquely Kr -saturated?(with Hartke)
Derrick Stolee (UNL) Computational Combinatorics 8 / 68
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Overview The Big Question
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 8 / 68
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Overview The Big Question
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 8 / 68
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Overview Combinatorial Search
Main Technique: Combinatorial Search
Goal: Determine if certain combinatorial objects exist with givenstructural or extremal properties.
Idea: Build objects piece-by-piece from base examples toenumerate all desired examples of a given order.
Most interesting properties are invariant under isomorphism.
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Overview Combinatorial Search
Main Technique: Combinatorial Search
Goal: Determine if certain combinatorial objects exist with givenstructural or extremal properties.
Idea: Build objects piece-by-piece from base examples toenumerate all desired examples of a given order.
Most interesting properties are invariant under isomorphism.
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Overview Combinatorial Search
Combinatorial Object: Graphs
A graph G of order n is composed of a set V (G) of n vertices and aset E(G) of edges, where the edges are unordered pairs of vertices.
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Overview Combinatorial Search
Combinatorial Object: Graphs
A graph G of order n is composed of a set V (G) of n vertices and aset E(G) of edges, where the edges are unordered pairs of vertices.
1
2
34
5 67
89
10
23
4
567
8
9
10
1 1
2
34
5 6
7
89
10
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Overview Combinatorial Search
Combinatorial Object: Graphs
An isomorphism between G1 and G2 is a bijection from V (G1) toV (G2) that induces a bijection from E(G1) to E(G2).
1
2
34
5 67
89
10
23
4
567
8
9
10
1 1
2
34
5 6
7
89
10
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
Derrick Stolee (UNL) Computational Combinatorics 11 / 68
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
Derrick Stolee (UNL) Computational Combinatorics 11 / 68
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
Derrick Stolee (UNL) Computational Combinatorics 11 / 68
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
Derrick Stolee (UNL) Computational Combinatorics 11 / 68
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Overview Combinatorial Search
Example: Generating Graphs by Edges
We can build graphs starting at Kn by adding edges.
Derrick Stolee (UNL) Computational Combinatorics 11 / 68
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Overview Combinatorial Search
Two Techniques for Isomorphs
1 Canonical Deletion (McKay 1998)
Removes all isomorphs.Not known how to integrate with constraint propagation.High cost per object.
Chapter 6
2 Orbital Branching (Ostrowski, Linderoth, Rossi, Smriglio 2007)Removes some, but not all isomorphs.Naturally integrates with constraint propagation.Low cost per object.
Chapter 10
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Overview Combinatorial Search
Two Techniques for Isomorphs
1 Canonical Deletion (McKay 1998)
Removes all isomorphs.Not known how to integrate with constraint propagation.High cost per object.
Overview in Chapter 6
2 Orbital Branching (Ostrowski, Linderoth, Rossi, Smriglio 2007)Removes some, but not all isomorphs.Naturally integrates with constraint propagation.Low cost per object.
Overview in Chapter 10
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
Derrick Stolee (UNL) Computational Combinatorics 13 / 68
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Search by Augmentations
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Overview Combinatorial Search
Implementation
My TreeSearch library enables parallelization in the Condor scheduler.
Executes on the Open Science Grid, a collection of supercomputersaround the country.
Derrick Stolee (UNL) Computational Combinatorics 14 / 68
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p-Extremal Graphs
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 15 / 68
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p-Extremal Graphs
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 15 / 68
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Derrick Stolee (UNL) Computational Combinatorics 16 / 68
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.Φ(G) is the number of perfect matchings in the graph G.
Derrick Stolee (UNL) Computational Combinatorics 16 / 68
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.Φ(G) is the number of perfect matchings in the graph G.
Derrick Stolee (UNL) Computational Combinatorics 16 / 68
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.Φ(G) is the number of perfect matchings in the graph G.
Φ(G) = 38 edges
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Φ(G) = 38 edges 11 edges
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Φ(G) = 38 edges 11 edges
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Φ(G) = 38 edges 11 edges
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Φ(G) = 3 Φ(G) = 38 edges 11 edges
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p-Extremal Graphs Perfect Matchings
Perfect Matchings
A perfect matching is a set of edges which cover each vertex exactlyonce.
Question (Dudek, Schmitt, 2010) What is the maximum number ofedges in a graph with exactly n vertices and p perfect matchings?
Definition Let n be an even number and fix p ≥ 1.
f (n, p) = max{|E(G)| : |V (G)| = n, Φ(G) = p}.
Graphs attaining this number of edges are p-extremal.
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p-Extremal Graphs Perfect Matchings
Hetyei’s Theorem
Theorem (Hetyei’s Theorem, 1986) For all even n ≥ 2,
f (n, 1) =n2
4.
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p-Extremal Graphs Perfect Matchings
Hetyei’s Theorem
Theorem (Hetyei’s Theorem, 1986) For all even n ≥ 2,
f (n, 1) =n2
4.
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p-Extremal Graphs Perfect Matchings
Hetyei’s Theorem
Theorem (Hetyei’s Theorem, 1986) For all even n ≥ 2,
f (n, 1) =n2
4.
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p-Extremal Graphs Perfect Matchings
Hetyei’s Theorem
Theorem (Hetyei’s Theorem, 1986) For all even n ≥ 2,
f (n, 1) =n2
4.
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p-Extremal Graphs Perfect Matchings
The Form of f (n, p)
Theorem (Dudek, Schmitt, 2010) For each p, there exist constants
np, cp so that for all n ≥ np,
f (n, p) =n2
4+ cp.
p 1 2 3 4 5 6cp 0 1 2 2 2 3
H Dudek, Schmitt, 2010
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p-Extremal Graphs Perfect Matchings
The Form of f (n, p)
Theorem (Dudek, Schmitt, 2010) For each p, there exist constants
np, cp so that for all n ≥ np,
f (n, p) =n2
4+ cp.
p 1 2 3 4 5 6cp 0 1 2 2 2 3
H Dudek, Schmitt, 2010
Derrick Stolee (UNL) Computational Combinatorics 18 / 68
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p-Extremal Graphs Perfect Matchings
Structure Theorem
Theorem (Hartke, Stolee, West, Yancey, 2011) For a fixed p, everygraph G with n vertices, p perfect matchings, and f (n, p) = n2
4 + cpedges is composed of a finite list of fundamental graphs combined inspecified ways.
Proof involves several classic structure theorems from matching theoryin an extremal setting.
For p ≤ 10, the graphs have order at most 12.
Using standard software (McKay’s geng) we found the graphs andcomputed cp.
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p-Extremal Graphs Perfect Matchings
Structure Theorem
Theorem (Hartke, Stolee, West, Yancey, 2011) For a fixed p, everygraph G with n vertices, p perfect matchings, and f (n, p) = n2
4 + cpedges is composed of a finite list of fundamental graphs combined inspecified ways.
Proof involves several classic structure theorems from matching theoryin an extremal setting.
For p ≤ 10, the graphs have order at most 12.
Using standard software (McKay’s geng) we found the graphs andcomputed cp.
Derrick Stolee (UNL) Computational Combinatorics 19 / 68
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p-Extremal Graphs Perfect Matchings
Structure Theorem
Theorem (Hartke, Stolee, West, Yancey, 2011) For a fixed p, everygraph G with n vertices, p perfect matchings, and f (n, p) = n2
4 + cpedges is composed of a finite list of fundamental graphs combined inspecified ways.
Proof involves several classic structure theorems from matching theoryin an extremal setting.
For p ≤ 10, the graphs have order at most 12.
Using standard software (McKay’s geng) we found the graphs andcomputed cp.
Derrick Stolee (UNL) Computational Combinatorics 19 / 68
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p-Extremal Graphs Perfect Matchings
Fundamental Graphs for 2 ≤ p ≤ 10
p = 2 p = 3 p = 4 p = 5 p = 5
p = 6 p = 6 p = 7 p = 8 p = 8
p = 8 p = 9 p = 10Derrick Stolee (UNL) Computational Combinatorics 20 / 68
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p-Extremal Graphs Perfect Matchings
cp for small p
p 1 2 3 4 5 6 7 8 9 10cp 0 1 2 2 2 3 3 3 4 4
H Dudek, Schmitt 2010 HSWY 2011
Q: Is cp monotone in p?
Derrick Stolee (UNL) Computational Combinatorics 21 / 68
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p-Extremal Graphs Perfect Matchings
cp for small p
p 1 2 3 4 5 6 7 8 9 10cp 0 1 2 2 2 3 3 3 4 4
H Dudek, Schmitt 2010 HSWY 2011
Q: Is cp monotone in p?
Derrick Stolee (UNL) Computational Combinatorics 21 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Structural Theorem, Redux
Without more involved computational methods, brute force methods(such as geng) cannot go farther.
The Lovasz Two Ear Theorem (1983) provides a way to build funda-mental graphs using ear augmentations.
Derrick Stolee (UNL) Computational Combinatorics 22 / 68
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p-Extremal Graphs Perfect Matchings
Computational Method
Developed a computational method from:
1. Augmentations: Lovasz Two Ear Theorem.
2. Isomorphs: Canonical Deletion. McKay
3. Pruning: Developed new structural and extremal theorems.
Before: Stuck at p ≤ 10 when searching on most 12 vertices.
Now: Found graphs for all p ≤ 27 on up to 22 vertices.
Derrick Stolee (UNL) Computational Combinatorics 23 / 68
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p-Extremal Graphs Perfect Matchings
Computational Method
Developed a computational method from:
1. Augmentations: Lovasz Two Ear Theorem.
2. Isomorphs: Canonical Deletion. McKay
3. Pruning: Developed new structural and extremal theorems.
Before: Stuck at p ≤ 10 when searching on most 12 vertices.
Now: Found graphs for all p ≤ 27 on up to 22 vertices.
Derrick Stolee (UNL) Computational Combinatorics 23 / 68
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p-Extremal Graphs Perfect Matchings
Computational Method
Developed a computational method from:
1. Augmentations: Lovasz Two Ear Theorem.
2. Isomorphs: Canonical Deletion. McKay
3. Pruning: Developed new structural and extremal theorems.
Before: Stuck at p ≤ 10 when searching on most 12 vertices.
Now: Found graphs for all p ≤ 27 on up to 22 vertices.
Derrick Stolee (UNL) Computational Combinatorics 23 / 68
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p-Extremal Graphs Perfect Matchings
Fundamental Graphs for 11 ≤ p ≤ 27
p = 11 p = 11 p = 12 p = 13 p = 13 p = 13 p = 13 p = 13
p = 13 p = 14 p = 14 p = 15 p = 16 p = 16 p = 16 p = 16
p = 17 p = 17 p = 18 p = 18 p = 19 p = 19 p = 19 p = 19
p = 19 p = 19 p = 20 p = 21 p = 21 p = 21 p = 22 p = 23
p = 24 p = 24 p = 25 p = 25 p = 26 p = 26 p = 26 p = 27
Derrick Stolee (UNL) Computational Combinatorics 24 / 68
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p-Extremal Graphs Perfect Matchings
cp for small pp 1 2 3 4 5 6 7 8 9 10
cp 0 1 2 2 2 3 3 3 4 4H Dudek, Schmitt, 2010 HSWY, 2011
p 11 12 13 14 15 16 17 18 19 20cp 3 5 3 4 6 4 4 5 4 5
Stolee, 2011
p 21 22 23 24 25 26 27cp 5 5 5 6 5 5 6
Stolee, 2011
Derrick Stolee (UNL) Computational Combinatorics 25 / 68
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p-Extremal Graphs Perfect Matchings
cp for small pp 1 2 3 4 5 6 7 8 9 10
cp 0 1 2 2 2 3 3 3 4 4H Dudek, Schmitt, 2010 HSWY, 2011
p 11 12 13 14 15 16 17 18 19 20cp 3 5 3 4 6 4 4 5 4 5
Stolee, 2011
p 21 22 23 24 25 26 27cp 5 5 5 6 5 5 6
Stolee, 2011
cp not monotonic in p !Blue numbers match conjectured upper bound.
Derrick Stolee (UNL) Computational Combinatorics 25 / 68
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Uniquely Kr -Saturated Graphs
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 26 / 68
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Uniquely Kr -Saturated Graphs
Problems Tackled in This Thesis
1 Which numbers are representable as the number of chains in awidth-two poset?(with Kupin, Reiniger) Chapter 4
2 Which colorings of {1, . . . , n} avoid monochromatic progressions?(with Jobson, Kezdy) Chapter 5
3 How many edges can exist in a graph with p perfect matchings?(with Hartke, West, Yancey) Chapter 9
4 What graphs are uniquely Kr -saturated?(with Hartke) Chapter 11
Derrick Stolee (UNL) Computational Combinatorics 26 / 68
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Uniquely Kr -Saturated Graphs H-Saturated Graphs
H-Saturated Graphs
Definition A graph G is H-saturated if
◦ G does not contain H as a subgraph. (H-free)◦ For every e ∈ E(G), G + e contains H as a subgraph.
5-cycle 6-cycle
Example: H = K3 where Kr is the complete graph on r vertices.Derrick Stolee (UNL) Computational Combinatorics 27 / 68
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Uniquely Kr -Saturated Graphs H-Saturated Graphs
H-Saturated Graphs
Definition A graph G is H-saturated if
◦ G does not contain H as a subgraph. (H-free)◦ For every e ∈ E(G), G + e contains H as a subgraph.
5-cycle 6-cycleis K3-saturated
Example: H = K3 where Kr is the complete graph on r vertices.Derrick Stolee (UNL) Computational Combinatorics 27 / 68
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Uniquely Kr -Saturated Graphs H-Saturated Graphs
H-Saturated Graphs
Definition A graph G is H-saturated if
◦ G does not contain H as a subgraph. (H-free)◦ For every e ∈ E(G), G + e contains H as a subgraph.
5-cycle 6-cycleis K3-saturated
Example: H = K3 where Kr is the complete graph on r vertices.Derrick Stolee (UNL) Computational Combinatorics 27 / 68
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Uniquely Kr -Saturated Graphs H-Saturated Graphs
H-Saturated Graphs
Definition A graph G is H-saturated if
◦ G does not contain H as a subgraph. (H-free)◦ For every e ∈ E(G), G + e contains H as a subgraph.
5-cycle 6-cycleis K3-saturated is not K3-saturated
Example: H = K3 where Kr is the complete graph on r vertices.Derrick Stolee (UNL) Computational Combinatorics 27 / 68
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Uniquely Kr -Saturated Graphs H-Saturated Graphs
Turan’s Theorem
Theorem (Turan, 1941) Let r ≥ 3. If G is Kr -saturated on n vertices,then G has at most
(1− 1
r−1
) n2
2 edges (asymptotically).
Derrick Stolee (UNL) Computational Combinatorics 28 / 68
Page 88
Uniquely Kr -Saturated Graphs H-Saturated Graphs
Turan’s Theorem
Theorem (Turan, 1941) Let r ≥ 3. If G is Kr -saturated on n vertices,then G has at most
(1− 1
r−1
) n2
2 edges (asymptotically).
r − 1 parts
Derrick Stolee (UNL) Computational Combinatorics 28 / 68
Page 89
Uniquely Kr -Saturated Graphs H-Saturated Graphs
Turan’s Theorem
Theorem (Turan, 1941) Let r ≥ 3. If G is Kr -saturated on n vertices,then G has at most
(1− 1
r−1
) n2
2 edges (asymptotically).
r − 1 parts Many copies of Kr !
Derrick Stolee (UNL) Computational Combinatorics 28 / 68
Page 90
Uniquely Kr -Saturated Graphs H-Saturated Graphs
Erdos, Hajnal, and Moon
Theorem (Erdos, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr -saturatedon n vertices, then G has at least (r−2
2 ) + (r − 2)(n− r + 2) edges.
Derrick Stolee (UNL) Computational Combinatorics 29 / 68
Page 91
Uniquely Kr -Saturated Graphs H-Saturated Graphs
Erdos, Hajnal, and Moon
Theorem (Erdos, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr -saturatedon n vertices, then G has at least (r−2
2 ) + (r − 2)(n− r + 2) edges.
1-book 2-book 3-book
Derrick Stolee (UNL) Computational Combinatorics 29 / 68
Page 92
Uniquely Kr -Saturated Graphs H-Saturated Graphs
Erdos, Hajnal, and Moon
Theorem (Erdos, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr -saturatedon n vertices, then G has at least (r−2
2 ) + (r − 2)(n− r + 2) edges.
1-book 2-book 3-bookExactly one copy of Kr !
Derrick Stolee (UNL) Computational Combinatorics 29 / 68
Page 93
Uniquely H-Saturated Graphs Definition
Uniquely H-Saturated Graphs
The Turan graph has many copies of Kr when an edge is added.
The books have exactly one copy of Kr when an edge is added.
Definition A graph G is uniquely H-saturated if G does not containH as a subgraph and for every edge e ∈ G admits exactly one copy ofH in G + e.
We consider the case where H = Kr (an r -clique).
Derrick Stolee (UNL) Computational Combinatorics 30 / 68
Page 94
Uniquely H-Saturated Graphs Definition
Uniquely H-Saturated Graphs
The Turan graph has many copies of Kr when an edge is added.
The books have exactly one copy of Kr when an edge is added.
Definition A graph G is uniquely H-saturated if G does not containH as a subgraph and for every edge e ∈ G admits exactly one copy ofH in G + e.
We consider the case where H = Kr (an r -clique).
Derrick Stolee (UNL) Computational Combinatorics 30 / 68
Page 95
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Uniquely K3-Saturated Graphs
Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011)The uniquely K3-saturated graphs are either stars or Moore graphs ofdiameter 2 and girth 5.
Theorem (Hoffman, Singleton, 1964) There are a finite number ofMoore graphs of diameter 2 and girth 5.
?C5 Petersen Hoffman– 57-Regular
Singleton Order 3250
Derrick Stolee (UNL) Computational Combinatorics 31 / 68
Page 96
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Uniquely K3-Saturated Graphs
Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011)The uniquely K3-saturated graphs are either stars or Moore graphs ofdiameter 2 and girth 5.
Theorem (Hoffman, Singleton, 1964) There are a finite number ofMoore graphs of diameter 2 and girth 5.
?C5 Petersen Hoffman– 57-Regular
Singleton Order 3250
Derrick Stolee (UNL) Computational Combinatorics 31 / 68
Page 97
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Uniquely K3-Saturated Graphs
Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011)The uniquely K3-saturated graphs are either stars or Moore graphs ofdiameter 2 and girth 5.
Theorem (Hoffman, Singleton, 1964) There are a finite number ofMoore graphs of diameter 2 and girth 5.
?C5 Petersen Hoffman– 57-Regular
Singleton Order 3250
Derrick Stolee (UNL) Computational Combinatorics 31 / 68
Page 98
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Dominating Vertices
Adding a dominating vertex to a uniquely Kr -saturated graph creates auniquely Kr+1-saturated graph.
Call uniquely Kr -saturated graphs without a dominating vertex
r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 32 / 68
Page 99
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Dominating Vertices
Adding a dominating vertex to a uniquely Kr -saturated graph creates auniquely Kr+1-saturated graph.
Call uniquely Kr -saturated graphs without a dominating vertex
r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 32 / 68
Page 100
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Dominating Vertices
Adding a dominating vertex to a uniquely Kr -saturated graph creates auniquely Kr+1-saturated graph.
Call uniquely Kr -saturated graphs without a dominating vertex
r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 32 / 68
Page 101
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Dominating Vertices
Adding a dominating vertex to a uniquely Kr -saturated graph creates auniquely Kr+1-saturated graph.
Call uniquely Kr -saturated graphs without a dominating vertex
r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 32 / 68
Page 102
Uniquely H-Saturated Graphs Uniquely K3-Saturated Graphs
Dominating Vertices
Adding a dominating vertex to a uniquely Kr -saturated graph creates auniquely Kr+1-saturated graph.
Call uniquely Kr -saturated graphs without a dominating vertex
r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 32 / 68
Page 103
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
2-primitive graphs are empty graphs.
3-primitive graphs are Moore graphs of diameter 2 and girth 5.
Derrick Stolee (UNL) Computational Combinatorics 33 / 68
Page 104
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
2-primitive graphs are empty graphs.
3-primitive graphs are Moore graphs of diameter 2 and girth 5.
Derrick Stolee (UNL) Computational Combinatorics 33 / 68
Page 105
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
2-primitive graphs are empty graphs.
3-primitive graphs are Moore graphs of diameter 2 and girth 5.
Derrick Stolee (UNL) Computational Combinatorics 33 / 68
Page 106
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
For r ≥ 1, C2r−1 is r -primitive.
C5 C7 C9
(Collins, Cooper, Kay, Wenger, 2010)
Derrick Stolee (UNL) Computational Combinatorics 34 / 68
Page 107
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
For r ≥ 1, C2r−1 is r -primitive.
C5 C7 C9
(Collins, Cooper, Kay, Wenger, 2010)
Derrick Stolee (UNL) Computational Combinatorics 34 / 68
Page 108
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
For r ≥ 1, C2r−1 is r -primitive.
C5 C7 C9
(Collins, Cooper, Kay, Wenger, 2010)
Derrick Stolee (UNL) Computational Combinatorics 34 / 68
Page 109
Uniquely H-Saturated Graphs Known r -Primitive Graphs
r -Primitive Graphs
A uniquely Kr -saturated graph with no dominating vertex is r -primitive.
For r ≥ 1, C2r−1 is r -primitive.
C5 C7 C9
(Collins, Cooper, Kay, Wenger, 2010)
Derrick Stolee (UNL) Computational Combinatorics 34 / 68
Page 110
Uniquely H-Saturated Graphs Known r -Primitive Graphs
Uniquely K4-Saturated Graphs
10 vertices 12 vertices
Previously known 4-primitive graphs (Collins, Cooper, Kay, 2010)
Derrick Stolee (UNL) Computational Combinatorics 35 / 68
Page 111
Uniquely H-Saturated Graphs Main Questions
Two Questions
1. Fix r ≥ 3. Are there a finite number of r -primitive graphs?
2. Is every r -primitive graph regular?
Derrick Stolee (UNL) Computational Combinatorics 36 / 68
Page 112
Uniquely H-Saturated Graphs Main Questions
Two Questions
1. Fix r ≥ 3. Are there a finite number of r -primitive graphs?
2. Is every r -primitive graph regular?
Derrick Stolee (UNL) Computational Combinatorics 36 / 68
Page 113
Uniquely H-Saturated Graphs Main Questions
Two Questions
1. Fix r ≥ 3. Are there a finite number of r -primitive graphs?
2. Is every r -primitive graph regular?
Derrick Stolee (UNL) Computational Combinatorics 36 / 68
Page 114
Uniquely H-Saturated Graphs Computational Method
Edges and Non-Edges
Non-edges are crucial to the structure of r -primitive graphs.
Derrick Stolee (UNL) Computational Combinatorics 37 / 68
Page 115
Uniquely H-Saturated Graphs Computational Method
Edges and Non-Edges
Non-edges are crucial to the structure of r -primitive graphs.
Edge Non-edge
Unassigned
TrigraphDerrick Stolee (UNL) Computational Combinatorics 37 / 68
Page 116
Uniquely H-Saturated Graphs Computational Method
Kr -Completions
For every non-edge we add, we add a Kr -completion:
ij a non-edge if and only if there exists a set S ⊂ [n], |S| = r − 2,so that ia, ja, and ab are edges for all a, b ∈ S.
S S S
r = 4 r = 5 r = 6
Derrick Stolee (UNL) Computational Combinatorics 38 / 68
Page 117
Uniquely H-Saturated Graphs Computational Method
Computational Method
Developed a computational method from:
1. Augmentations: Kr -Completions.
2. Isomorphs: Orbital Branching. Ostrowsky et al.
3. Pruning: Contains Kr or double-completion.
Derrick Stolee (UNL) Computational Combinatorics 39 / 68
Page 118
Uniquely H-Saturated Graphs Computational Method
Exhaustive Search Times
n r = 4 r = 5 r = 6 r = 7 r = 810 0.10 s 0.37 s 0.13 s 0.01 s 0.01 s11 0.68 s 5.25 s 1.91 s 0.28 s 0.09 s12 4.58 s 1.60 m 25.39 s 1.97 s 1.12 s13 34.66 s 34.54 m 6.53 m 59.94 s 20.03 s14 4.93 m 10.39 h 5.13 h 20.66 m 2.71 m15 40.59 m 23.49 d 10.08 d 12.28 h 1.22 h16 6.34 h 1.58 y 1.74 y 34.53 d 1.88 d17 3.44 d 8.76 y 115.69 d18 53.01 d19 2.01 y20 45.11 y
Total CPU times using Open Science Grid.
Derrick Stolee (UNL) Computational Combinatorics 40 / 68
Page 119
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Page 120
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Empty graphs
Page 121
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Empty graphs
Cycle complements
Page 122
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Empty graphs
Cycle complements
Old examples
Page 123
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Empty graphs
Cycle complements
Old examples
New examples
Page 124
←− clique size−→
←−
vert
ices−→
n \ r 2 3 4 5 6 7 82345678910111213141516171819
Empty graphs
Cycle complements
Old examples
New examples
Page 125
Uniquely H-Saturated Graphs 4-Primitive Graphs
4-Primitive Graphsn = 13
G(A)13 Paley(13)
Derrick Stolee (UNL) Computational Combinatorics 42 / 68
Page 126
Uniquely H-Saturated Graphs 5-Primitive Graph
5-Primitive Graphn = 16 : G(A)
16
Derrick Stolee (UNL) Computational Combinatorics 43 / 68
Page 127
Uniquely H-Saturated Graphs 5-Primitive Graph
5-Primitive Graphn = 16 : G(A)
16
Derrick Stolee (UNL) Computational Combinatorics 43 / 68
Page 128
Uniquely H-Saturated Graphs 5-Primitive Graph
5-Primitive Graphn = 16 : G(A)
16
Not all r -primitive graphs are regular!
Derrick Stolee (UNL) Computational Combinatorics 43 / 68
Page 129
Uniquely H-Saturated Graphs 5-Primitive Graph
7-Primitive Graphn = 17 : G(A)
17
Derrick Stolee (UNL) Computational Combinatorics 44 / 68
Page 130
Uniquely H-Saturated Graphs 5-Primitive Graph
7-Primitive Graphn = 17 : G(A)
17
Derrick Stolee (UNL) Computational Combinatorics 44 / 68
Page 131
Uniquely H-Saturated Graphs Cayley Complements
Let Γ be a group and S ⊆ Γ a set of generators.
The undirected Cayley graph C(Γ, S) has vertex set Γ and for alla ∈ Γ and b ∈ S, there is an edge between a and ab.
The Cayley complement C(Γ, S) is the complement of C(Γ, S).
For r ≥ 1, C(Z2r−1, {1}) ∼= C2r−1 is r -primitive.
C5 C7 C9
Derrick Stolee (UNL) Computational Combinatorics 45 / 68
Page 132
Uniquely H-Saturated Graphs Cayley Complements
Let Γ be a group and S ⊆ Γ a set of generators.
The undirected Cayley graph C(Γ, S) has vertex set Γ and for alla ∈ Γ and b ∈ S, there is an edge between a and ab.
The Cayley complement C(Γ, S) is the complement of C(Γ, S).
For r ≥ 1, C(Z2r−1, {1}) ∼= C2r−1 is r -primitive.
C5 C7 C9
Derrick Stolee (UNL) Computational Combinatorics 45 / 68
Page 133
Uniquely H-Saturated Graphs Cayley Complements
Let Γ be a group and S ⊆ Γ a set of generators.
The undirected Cayley graph C(Γ, S) has vertex set Γ and for alla ∈ Γ and b ∈ S, there is an edge between a and ab.
The Cayley complement C(Γ, S) is the complement of C(Γ, S).
For r ≥ 1, C(Z2r−1, {1}) ∼= C2r−1 is r -primitive.
C5 C7 C9
Derrick Stolee (UNL) Computational Combinatorics 45 / 68
Page 134
Uniquely H-Saturated Graphs Cayley Complements
Two or Three Generators
S r n{1, 4} 7 17
{1, 6} 16 37
{1, 8} 29 65
{1, 10} 46 101
{1, 12} 67 145
g = 2
S r n{1, 5, 6} 9 31
{1, 8, 9} 22 73
{1, 11, 12} 41 133
{1, 14, 15} 66 211
{1, 17, 18} 97 307
g = 3
Derrick Stolee (UNL) Computational Combinatorics 46 / 68
Page 135
Uniquely H-Saturated Graphs Cayley Complements
Infinite Families
Conjecture (Hartke, Stolee, 2012) Let t ≥ 1,
n = 4t2 + 1, and r = 2t2 − t + 1.
The Cayley complement C(Zn, {1, 2t}) is r -primitive.
Conjecture (Hartke, Stolee, 2012) Let t ≥ 1,
n = 9t2 − 3t + 1 and r = 3t2 − 2t + 1.
The Cayley complement C(Zn, {1, 3t − 1, 3t}) is r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 47 / 68
Page 136
Uniquely H-Saturated Graphs Cayley Complements
Infinite Families
Theorem (Hartke, Stolee, 2012) Let t ≥ 1,
n = 4t2 + 1, and r = 2t2 − t + 1.
The Cayley complement C(Zn, {1, 2t}) is r -primitive.
Proof uses counting method.
Conjecture (Hartke, Stolee, 2012) Let t ≥ 1,
n = 9t2 − 3t + 1 and r = 3t2 − 2t + 1.
The Cayley complement C(Zn, {1, 3t − 1, 3t}) is r -primitive.
Derrick Stolee (UNL) Computational Combinatorics 47 / 68
Page 137
Uniquely H-Saturated Graphs Cayley Complements
Infinite Families
Theorem (Hartke, Stolee, 2012) Let t ≥ 1,
n = 4t2 + 1, and r = 2t2 − t + 1.
The Cayley complement C(Zn, {1, 2t}) is r -primitive.
Proof uses counting method.
Theorem (Hartke, Stolee, 2012) Let t ≥ 1,
n = 9t2 − 3t + 1 and r = 3t2 − 2t + 1.
The Cayley complement C(Zn, {1, 3t − 1, 3t}) is r -primitive.
Proof uses discharging method.
Derrick Stolee (UNL) Computational Combinatorics 47 / 68
Page 138
Computational Combinatorics
PureCombinatorics
Algorithmsand Computation
Computational Combinatorics
Page 142
Space-Bounded Complexity
Complexity Results in This Thesis
1 ReachFewL = ReachUL.(with Garvin, Tewari, Vinodchandran) Chapter 13
2 Reachability in surface-embedded acyclic digraphs.(with Vinodchandran) Chapter 14
Derrick Stolee (UNL) Computational Combinatorics 52 / 68
Page 143
Space-Bounded Complexity Space-Bounded Complexity
Space-Bounded Complexity
A language is in L if there is a deterministic Turing machine thatdecides the language using at most O(log(n)) work cells.
A language is in NL if there is a non-deterministic Turing machinethat decides the language using at most O(log(n)) work cells.
L ⊆ NL
Derrick Stolee (UNL) Computational Combinatorics 53 / 68
Page 144
Space-Bounded Complexity Space-Bounded Complexity
Space-Bounded Complexity
A language is in L if there is a deterministic Turing machine thatdecides the language using at most O(log(n)) work cells.
A language is in NL if there is a non-deterministic Turing machinethat decides the language using at most O(log(n)) work cells.
L ⊆ NL
Derrick Stolee (UNL) Computational Combinatorics 53 / 68
Page 145
Space-Bounded Complexity Space-Bounded Complexity
Configuration Graphs
If M is an O(log(n))-space non-deterministic Turing machine andx ∈ {0, 1}∗, the configuration graph GM,x has
1 Vertices are configurations: assignments of state, work cellcontents, and tape head positions.
(Requires O(log n) bits to describe.)2 An edge C → C ′ exists if there is a transition function of M whose
operation on C results in C ′.
M accepts x if and only if there is a path from Cinit to Caccept in GM,x.
GM,x has poly-size and can be written using log-space.
Derrick Stolee (UNL) Computational Combinatorics 54 / 68
Page 146
Space-Bounded Complexity Space-Bounded Complexity
Configuration Graphs
If M is an O(log(n))-space non-deterministic Turing machine andx ∈ {0, 1}∗, the configuration graph GM,x has
1 Vertices are configurations: assignments of state, work cellcontents, and tape head positions.
(Requires O(log n) bits to describe.)
2 An edge C → C ′ exists if there is a transition function of M whoseoperation on C results in C ′.
M accepts x if and only if there is a path from Cinit to Caccept in GM,x.
GM,x has poly-size and can be written using log-space.
Derrick Stolee (UNL) Computational Combinatorics 54 / 68
Page 147
Space-Bounded Complexity Space-Bounded Complexity
Configuration Graphs
If M is an O(log(n))-space non-deterministic Turing machine andx ∈ {0, 1}∗, the configuration graph GM,x has
1 Vertices are configurations: assignments of state, work cellcontents, and tape head positions.
(Requires O(log n) bits to describe.)2 An edge C → C ′ exists if there is a transition function of M whose
operation on C results in C ′.
M accepts x if and only if there is a path from Cinit to Caccept in GM,x.
GM,x has poly-size and can be written using log-space.
Derrick Stolee (UNL) Computational Combinatorics 54 / 68
Page 148
Space-Bounded Complexity Space-Bounded Complexity
Configuration Graphs
If M is an O(log(n))-space non-deterministic Turing machine andx ∈ {0, 1}∗, the configuration graph GM,x has
1 Vertices are configurations: assignments of state, work cellcontents, and tape head positions.
(Requires O(log n) bits to describe.)2 An edge C → C ′ exists if there is a transition function of M whose
operation on C results in C ′.
M accepts x if and only if there is a path from Cinit to Caccept in GM,x.
GM,x has poly-size and can be written using log-space.
Derrick Stolee (UNL) Computational Combinatorics 54 / 68
Page 149
Space-Bounded Complexity Space-Bounded Complexity
Configuration Graphs
If M is an O(log(n))-space non-deterministic Turing machine andx ∈ {0, 1}∗, the configuration graph GM,x has
1 Vertices are configurations: assignments of state, work cellcontents, and tape head positions.
(Requires O(log n) bits to describe.)2 An edge C → C ′ exists if there is a transition function of M whose
operation on C results in C ′.
M accepts x if and only if there is a path from Cinit to Caccept in GM,x.
GM,x has poly-size and can be written using log-space.
Derrick Stolee (UNL) Computational Combinatorics 54 / 68
Page 150
Space-Bounded Complexity Space-Bounded Complexity
Meta-Theory of Space-Bounded Complexity
Every space-bounded complexity problem can bereduced to some form of the reachability problem indigraphs.
Reach = {〈G, s, t〉 : G is a directed graph with a path from s to t}
L ⊆ NL ⊆ P
Derrick Stolee (UNL) Computational Combinatorics 55 / 68
Page 151
Space-Bounded Complexity Space-Bounded Complexity
Meta-Theory of Space-Bounded Complexity
Every space-bounded complexity problem can bereduced to some form of the reachability problem indigraphs.
Reach = {〈G, s, t〉 : G is a directed graph with a path from s to t}
L ⊆ NL ⊆ P
Derrick Stolee (UNL) Computational Combinatorics 55 / 68
Page 152
Space-Bounded Complexity Space-Bounded Complexity
Complexity Results in This Thesis
1 ReachFewL = ReachUL.(with Garvin, Tewari, Vinodchandran) Chapter 13
2 Reachability in surface-embedded acyclic digraphs.(with Vinodchandran) Chapter 14
Derrick Stolee (UNL) Computational Combinatorics 56 / 68
Page 153
Space-Bounded Complexity Space-Bounded Complexity
Complexity Results in This Thesis
1 ReachFewL = ReachUL.(with Garvin, Tewari, Vinodchandran) Chapter 13
2 Reachability in surface-embedded acyclic digraphs.(with Vinodchandran) Chapter 14
Derrick Stolee (UNL) Computational Combinatorics 56 / 68
Page 154
Space-Bounded Complexity Space-Bounded Complexity
Log-space Classes and Reachability
LDeterministic
Complete:Undirected Reach
(Reingold 08)
Derrick Stolee (UNL) Computational Combinatorics 57 / 68
Page 155
Space-Bounded Complexity Space-Bounded Complexity
Log-space Classes and Reachability
LDeterministic
Complete:Undirected Reach
(Reingold 08)
NLNondeterministic
Complete:Directed Reach
Derrick Stolee (UNL) Computational Combinatorics 57 / 68
Page 156
Space-Bounded Complexity Space-Bounded Complexity
Log-space Classes and Reachability
LDeterministic
Complete:Undirected Reach
(Reingold 08)
ULUnambiguous
Contains:Dir. Planar Reach(Bourke, Tewari,
Vinodchandran 09)
NLNondeterministic
Complete:Directed Reach
Derrick Stolee (UNL) Computational Combinatorics 57 / 68
Page 157
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n] NL TISP[poly(n), n1−ε
]UL
OO
PlanarReach
L
OO
Derrick Stolee (UNL) Computational Combinatorics 58 / 68
Page 158
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n] NL TISP[poly(n), n1−ε
]UL
OO
PlanarReach
BTV ’09
OO
?��
L
Derrick Stolee (UNL) Computational Combinatorics 58 / 68
Page 159
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n] NL
Savitch, ’70hhPPPPPPPPPPPPPPPP
TISP[poly(n), n1−ε
]UL
OO
PlanarReach
BTV ’09
OO
?��
L
Derrick Stolee (UNL) Computational Combinatorics 58 / 68
Page 160
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n]
OO
NL
Savitch, ’70hhPPPPPPPPPPPPPPPP
TISP[poly(n), n1−ε
]UL
OO
PlanarReach
BTV ’09
OO
?
TT
?��
L
Derrick Stolee (UNL) Computational Combinatorics 58 / 68
Page 161
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n]
OO
NL
Savitch, ’70hhPPPPPPPPPPPPPPPP
BBRS, ’9266llllllllllllllll
TISP[poly(n), n1−ε
]UL
OO
PlanarReach
BTV ’09
OO
?
TT
?��
L
Derrick Stolee (UNL) Computational Combinatorics 58 / 68
Page 162
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Other Perspectives
SPACE[log2 n] TISP[poly(n), n
2√
log n
]
SPACE[log2−ε n]
OO
NL
Savitch, ’70hhPPPPPPPPPPPPPPPP
BBRS, ’9266llllllllllllllll
TISP[poly(n), n1−ε
]OO
UL
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BTV ’09
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Derrick Stolee (UNL) Computational Combinatorics 58 / 68
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar and Acyclic Restrictions
1 Reach for acyclic digraphs is complete for NL.
2 Reach for planar digraphs is in UL, but we believe UL = NL.
3 What if we combine acyclic and planar?
We also bound number of
sources sinks
Derrick Stolee (UNL) Computational Combinatorics 59 / 68
Page 164
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar and Acyclic Restrictions
1 Reach for acyclic digraphs is complete for NL.
2 Reach for planar digraphs is in UL, but we believe UL = NL.
3 What if we combine acyclic and planar?
We also bound number of
sources sinks
Derrick Stolee (UNL) Computational Combinatorics 59 / 68
Page 165
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar and Acyclic Restrictions
1 Reach for acyclic digraphs is complete for NL.
2 Reach for planar digraphs is in UL, but we believe UL = NL.
3 What if we combine acyclic and planar?
We also bound number of
sources sinks
Derrick Stolee (UNL) Computational Combinatorics 59 / 68
Page 166
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar and Acyclic Restrictions
1 Reach for acyclic digraphs is complete for NL.
2 Reach for planar digraphs is in UL, but we believe UL = NL.
3 What if we combine acyclic and planar?
We also bound number of
sources sinks
Derrick Stolee (UNL) Computational Combinatorics 59 / 68
Page 167
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar + Acyclic Reachability in Log-Space
1 Series-parallel graphs(Jakoby, Liskiewicz, Reischuk, Tantau, ’06/’07)
2 Single-source Single-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
3 Single-source Multiple-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
4 Log-source Multiple-Sink Planar DAGs(Stolee, Bourke, Vinodchandran, ’10)
Derrick Stolee (UNL) Computational Combinatorics 60 / 68
Page 168
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar + Acyclic Reachability in Log-Space
1 Series-parallel graphs(Jakoby, Liskiewicz, Reischuk, Tantau, ’06/’07)
2 Single-source Single-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
3 Single-source Multiple-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
4 Log-source Multiple-Sink Planar DAGs(Stolee, Bourke, Vinodchandran, ’10)
Derrick Stolee (UNL) Computational Combinatorics 60 / 68
Page 169
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar + Acyclic Reachability in Log-Space
1 Series-parallel graphs(Jakoby, Liskiewicz, Reischuk, Tantau, ’06/’07)
2 Single-source Single-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
3 Single-source Multiple-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
4 Log-source Multiple-Sink Planar DAGs(Stolee, Bourke, Vinodchandran, ’10)
Derrick Stolee (UNL) Computational Combinatorics 60 / 68
Page 170
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Planar + Acyclic Reachability in Log-Space
1 Series-parallel graphs(Jakoby, Liskiewicz, Reischuk, Tantau, ’06/’07)
2 Single-source Single-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
3 Single-source Multiple-Sink Planar DAGs(Allender, Barrington, Chakraborty, Datta, Roy, ’09)
4 Log-source Multiple-Sink Planar DAGs(Stolee, Bourke, Vinodchandran, ’10)
Derrick Stolee (UNL) Computational Combinatorics 60 / 68
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Surface-embedded graphs
We also extend to graphs embedded in surfaces of low genus.
Let G(m, g) denote the acyclic digraphs with m sources andembedded in a genus g surface.
Derrick Stolee (UNL) Computational Combinatorics 61 / 68
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Surface-embedded graphs
We also extend to graphs embedded in surfaces of low genus.
Let G(m, g) denote the acyclic digraphs with m sources andembedded in a genus g surface.
Derrick Stolee (UNL) Computational Combinatorics 61 / 68
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Reduction with Compression
Theorem (Stolee, Vinodchandran, ’12) Given a graph G ∈ G(m, g)and s, t ∈ V (G), we can compute in log-space a graph G′ withvertices s′, t ′ so that
1 There is a path from s to t in G if and only if there is a path froms′ to t ′ in G′.
2 G′ has O(m + g) vertices.
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Topological Equivalence
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Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Our Results (Stolee, Vinodchandran, ’12)
Theorem (Sub-Savitch) Reachability for graphs of order n in G(m, g)is in SPACE[log n + log2(m + g)].
Theorem (Log-Space) If m = g = 2√
log n, reach for G(m, g) is in L.
Theorem (Time-Space) Reachability for graphs of order n in G(m, g)is in TISP[poly(n), log n + m + g].
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Page 181
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Our Results (Stolee, Vinodchandran, ’12)
Theorem (Sub-Savitch) Reachability for graphs of order n in G(m, g)is in SPACE[log n + log2(m + g)].
Theorem (Log-Space) If m = g = 2√
log n, reach for G(m, g) is in L.
Theorem (Time-Space) Reachability for graphs of order n in G(m, g)is in TISP[poly(n), log n + m + g].
Derrick Stolee (UNL) Computational Combinatorics 64 / 68
Page 182
Space-Bounded Complexity Reachability in Surface-Embedded Graphs
Our Results (Stolee, Vinodchandran, ’12)
Theorem (Sub-Savitch) Reachability for graphs of order n in G(m, g)is in SPACE[log n + log2(m + g)].
Theorem (Log-Space) If m = g = 2√
log n, reach for G(m, g) is in L.
Theorem (Time-Space) Reachability for graphs of order n in G(m, g)is in TISP[poly(n), log n + m + g].
Derrick Stolee (UNL) Computational Combinatorics 64 / 68
Page 185
Combinatorics Using Computational Methods
Derrick Stolee
University of Nebraska–[email protected]
http://www.math.unl.edu/∼s-dstolee1/
March 13, 2012Dissertation Defense
Supported by NSF grants DMS-0354008, DMS-0914815, and CCF-0916525,
and a University of Nebraska Presidential Fellowship.