Computational Combinatorics and the search for Uniquely K r - - PowerPoint PPT Presentation

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Computational Combinatorics and the search for Uniquely K r - - PowerPoint PPT Presentation

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Computational Combinatorics and the search for Uniquely K r -Saturated Graphs Stephen G. Hartke University of NebraskaLincoln, USA hartke@math.unl.edu http://www.math.unl.edu/ shartke2/ Joint work with Derrick Stolee, Iowa State

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Computational Combinatorics and the search for Uniquely Kr-Saturated Graphs Stephen G. Hartke

University of Nebraska–Lincoln, USA hartke@math.unl.edu http://www.math.unl.edu/∼shartke2/

Joint work with Derrick Stolee, Iowa State University.

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Overview The Big Question

What is Computational Combinatorics?

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 2 / 1

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Overview The Big Question

What is Computational Combinatorics?

Using a combination of pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics by providing evidence for conjectures, finding examples and counterexamples, and discovering and proving theorems.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 3 / 1

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Overview The Big Question

The Goal

Determine if certain combinatorial objects exist with given structural or extremal properties.

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Overview The Big Question

The Goal

Determine if certain combinatorial objects exist with given structural 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, ¨ Osterg˚ ard, 2004)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 4 / 1

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Overview The Big Question

The Goal

Determine if certain combinatorial objects exist with given structural 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, ¨ Osterg˚ ard, 2004)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 4 / 1

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Overview The Big Question

Combinatorial Object: Graphs

A graph G of order n is composed of a set V(G) of n vertices and a set E(G) of edges, where the edges are unordered pairs of vertices.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 5 / 1

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Overview The Big Question

Combinatorial Object: Graphs

Cycles Ck C3 C4 C5 C6

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 6 / 1

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Overview The Big Question

Combinatorial Object: Graphs

Cycles Ck C3 C4 C5 C6 Complete Graphs Kr (cliques) K3 K4 K5 K6

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 6 / 1

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Overview Combinatorial Search

Main Technique: Combinatorial Search

Goal: Determine if certain combinatorial objects exist with given structural or extremal properties. Idea: Build objects piece-by-piece from base examples to enumerate all desired examples of a given order.

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Overview Combinatorial Search

Main Technique: Combinatorial Search

Goal: Determine if certain combinatorial objects exist with given structural or extremal properties. Idea: Build objects piece-by-piece from base examples to enumerate all desired examples of a given order. The computer performs a long, detailed case analysis.

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Overview Combinatorial Search

Main Technique: Combinatorial Search

Goal: Determine if certain combinatorial objects exist with given structural or extremal properties. Idea: Build objects piece-by-piece from base examples to enumerate all desired examples of a given order. The computer performs a long, detailed case analysis. Our job is to efficiently design the case analysis, using algorithms: Combinatorial Generation Combinatorial Optimization Graph Algorithms

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 7 / 1

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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.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

We can build graphs starting at Kn by adding edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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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.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

We can build graphs starting at Kn by adding edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

We can build graphs starting at Kn by adding edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

We can build graphs starting at Kn by adding edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

We can build graphs starting at Kn by adding edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 8 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

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Overview Combinatorial Search

Example: Generating Graphs by Edges

1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 9 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

An isomorphism between G1 and G2 is a bijection from V(G1) to V(G2) that induces a bijection from E(G1) to E(G2). 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 9 / 1

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Overview Combinatorial Search

Labeled Versus Unlabeled Objects

A labeled graph has a linear ordering on the vertices. An unlabeled graph represents an isomorphism class of graphs.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 10 / 1

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Overview Combinatorial Search

Labeled Versus Unlabeled Objects

A labeled graph has a linear ordering on the vertices. An unlabeled graph represents an isomorphism class of graphs. Most interesting graph properties are invariant under isomorphism.

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n

Labeled graphs of order n

6 32,768 7 2,097,152 8 268,435,456 9 68,719,476,736 10 35,184,372,088,832 11 36,028,797,018,963,968 12 73,786,976,294,838,206,464 13 302,231,454,903,657,293,676,544 14 2,475,880,078,570,760,549,798,248,448 15 40,564,819,207,303,340,847,894,502,572,032 2(n

2) ≈ 2θ(n2)

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n

Unlabeled connected graphs of order n

6 85 7 509 8 4,060 9 41,301 10 510,489 11 7,319,447 12 117,940,535 13 2,094,480,864 14 40,497,138,011 15 845,480,228,069 OEIS Sequence A002851 Grows 2Ω(n2).

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n

Unlabeled connected graphs of order n

6 85 7 509 8 4,060 9 41,301 10 510,489 11 7,319,447 12 117,940,535 13 2,094,480,864 14 40,497,138,011 15 845,480,228,069 Requires about 1 day of CPU Time.

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n

Unlabeled connected graphs of order n

6 85 7 509 8 4,060 9 41,301 10 510,489 11 7,319,447 12 117,940,535 13 2,094,480,864 14 40,497,138,011 15 845,480,228,069 Requires over 1 year of CPU Time.

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Overview Combinatorial Search

Shifting the Exponent

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 13 / 1

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Overview Combinatorial Search

Shifting the Exponent

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Overview Combinatorial Search

Shifting the Exponent

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 13 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

Unlabeled Graphs

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Overview Combinatorial Search

Example: Generating Graphs by Edges

Unlabeled Graphs

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 14 / 1

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Overview Combinatorial Search

Example: Generating Graphs by Edges

Unlabeled Graphs

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Overview Combinatorial Search

Example: Generating Graphs by Edges

Unlabeled Graphs

Multiple paths to same unlabeled

  • bject!

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 14 / 1

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Overview Search as a Poset

Toy Example

Suppose we are searching for graphs which are:

1

4-regular: All vertices have 4 incident edges.

2

3-colorable: The vertices can be colored with three colors so that no edge is monochromatic. 1 2 3 2 1 2 1 3 3 1 2 3 4 5

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 15 / 1

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Kn Kn

Unlabeled graphs

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Unlabeled graphs

4-regular, 3-chromatic graphs

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Sub-solutions

Unlabeled graphs

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Unlabeled graphs

Detectably not sub-solutions

∆(G) ≥ 5 χ(G) ≥ 4

Pruning

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Unlabeled graphs Ideal Path

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Unlabeled graphs Prune and Backtrack

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Unlabeled graphs Multiple paths!

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Unlabeled graphs Multiple paths!

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Unlabeled graphs Multiple paths!

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Unlabeled graphs Multiple paths!

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Unlabeled graphs Goal: Exactly one path.

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Unlabeled graphs Goal: Exactly one path.

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Unlabeled graphs Partition by subtrees.

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Parallelize!

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Overview Search as a Poset

Implementation

The TreeSearch library enables parallelization in the Condor scheduler. Executes on the Open Science Grid, a collection of supercomputers around the country.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 17 / 1

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Overview Search as a Poset

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 18 / 1

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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.

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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 is K3-saturated

Example: H = K3 where Kr is the complete graph on r vertices.

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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 is K3-saturated

Example: H = K3 where Kr is the complete graph on r vertices.

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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 is K3-saturated is not K3-saturated

Example: H = K3 where Kr is the complete graph on r vertices.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 19 / 1

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Tur´ an’s Theorem

Theorem (Tur´ an, 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).

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 20 / 1

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Tur´ an’s Theorem

Theorem (Tur´ an, 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

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 20 / 1

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Tur´ an’s Theorem

Theorem (Tur´ an, 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!

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 20 / 1

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Erd˝

  • s, Hajnal, and Moon

Theorem (Erd˝

  • s, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr-saturated
  • n n vertices, then G has at least (r−2

2 ) + (r − 2)(n − r + 2) edges.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 21 / 1

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Erd˝

  • s, Hajnal, and Moon

Theorem (Erd˝

  • s, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr-saturated
  • n n vertices, then G has at least (r−2

2 ) + (r − 2)(n − r + 2) edges.

1-book 2-book 3-book

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Uniquely Kr -Saturated Graphs H-Saturated Graphs

Erd˝

  • s, Hajnal, and Moon

Theorem (Erd˝

  • s, Hajnal, Moon, 1964) Let r ≥ 3. If G is Kr-saturated
  • n n vertices, then G has at least (r−2

2 ) + (r − 2)(n − r + 2) edges.

1-book 2-book 3-book Exactly one copy of Kr!

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Uniquely H-Saturated Graphs Definition

Uniquely H-Saturated Graphs

The Tur´ an graph has many copies of Kr when an edge is added. The books have exactly one copy of Kr when an edge is added.

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Uniquely H-Saturated Graphs Definition

Uniquely H-Saturated Graphs

The Tur´ an 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 contain H as a subgraph and for every edge e ∈ G admits exactly one copy of H in G + e.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 22 / 1

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C3-saturated graphs are either stars or Moore graphs of diameter 2 and girth 5.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 23 / 1

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C3-saturated graphs are either stars or Moore graphs of diameter 2 and girth 5. Theorem (Hoffman, Singleton, 1964) There are a finite number of Moore graphs of diameter 2 and girth 5.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 23 / 1

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C3-saturated graphs are either stars or Moore graphs of diameter 2 and girth 5. Theorem (Hoffman, Singleton, 1964) There are a finite number of Moore graphs of diameter 2 and girth 5.

?

C5 Petersen Hoffman– 57-Regular Singleton Order 3250

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C4-saturated graphs.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 24 / 1

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C4-saturated graphs. Theorem (Wenger, 2010) The only uniquely C5-saturated graphs are friendship graphs.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 24 / 1

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Uniquely H-Saturated Graphs Uniquely Ck -Saturated Graphs

Uniquely Ck-Saturated Graphs

Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C4-saturated graphs. Theorem (Wenger, 2010) The only uniquely C5-saturated graphs are friendship graphs. Theorem (Wenger, 2010) For k ∈ {6, 7, 8}, no uniquely Ck-saturated graph exists. Conjecture (Wenger, 2010) For k ≥ 9, no uniquely Ck-saturated graph exists.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 24 / 1

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Uniquely H-Saturated Graphs Definition

Uniquely Kr-Saturated Graphs

We consider the case where H = Kr (an r-clique) for r ≥ 4.

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Uniquely H-Saturated Graphs Definition

Uniquely Kr-Saturated Graphs

We consider the case where H = Kr (an r-clique) for r ≥ 4. (K3 ∼ = C3)

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Uniquely H-Saturated Graphs Definition

Uniquely Kr-Saturated Graphs

We consider the case where H = Kr (an r-clique) for r ≥ 4. (K3 ∼ = C3) Definition A graph G is uniquely Kr-saturated if G does not contain an r-clique and for every edge e ∈ G there is exactly one r-clique in G + e.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 25 / 1

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Uniquely H-Saturated Graphs Definition

Dominating Vertices

Adding a dominating vertex to a uniquely Kr-saturated graph creates a uniquely Kr+1-saturated graph.

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Uniquely H-Saturated Graphs Definition

Dominating Vertices

Adding a dominating vertex to a uniquely Kr-saturated graph creates a uniquely Kr+1-saturated graph.

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Uniquely H-Saturated Graphs Definition

Dominating Vertices

Adding a dominating vertex to a uniquely Kr-saturated graph creates a uniquely Kr+1-saturated graph.

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Uniquely H-Saturated Graphs Definition

Dominating Vertices

Adding a dominating vertex to a uniquely Kr-saturated graph creates a uniquely Kr+1-saturated graph.

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Uniquely H-Saturated Graphs Definition

Dominating Vertices

Adding a dominating vertex to a uniquely Kr-saturated graph creates a uniquely Kr+1-saturated graph. Call uniquely Kr-saturated graphs without a dominating vertex

r-primitive.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 26 / 1

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Uniquely H-Saturated Graphs Known r-Primitive Graphs

r-Primitive Graphs

A uniquely Kr-saturated graph with no dominating vertex is r-primitive.

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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.

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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.

?

C5 Petersen Hoffman– 57-Regular Singleton Order 3250

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 27 / 1

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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)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 28 / 1

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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)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 28 / 1

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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)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 28 / 1

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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)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 28 / 1

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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)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 29 / 1

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Uniquely H-Saturated Graphs Known r-Primitive Graphs

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch Problem

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 30 / 1

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Uniquely H-Saturated Graphs Main Questions

The Problem

Goal: Characterize uniquely Kr-saturated graphs. First Step: Reduce to r-primitive graphs.

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Uniquely H-Saturated Graphs Main Questions

The Problem

Goal: Characterize uniquely Kr-saturated graphs. First Step: Reduce to r-primitive graphs.

  • 1. Fix r ≥ 3. Are there a finite number of r-primitive graphs?

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 31 / 1

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Uniquely H-Saturated Graphs Main Questions

The Problem

Goal: Characterize uniquely Kr-saturated graphs. First Step: Reduce to r-primitive graphs.

  • 1. Fix r ≥ 3. Are there a finite number of r-primitive graphs?
  • 2. Is every r-primitive graph regular?

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 31 / 1

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Uniquely H-Saturated Graphs Computational Method

Edges and Non-Edges

Non-edges are crucial to the structure of r-primitive graphs.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 32 / 1

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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 Tricolored graph

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 32 / 1

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Uniquely H-Saturated Graphs Computational Method

Edges, Non-Edges, and Variables

Fix a vertex set {v1, v2, . . . , vn}. For i, j ∈ {1, . . . , n}, let xi,j =      1 vivj ∈ E(G) vivj / ∈ E(G) ∗ vivj unassigned . A vector x = (xi,j : i, j ∈ {1, . . . , n}) is a variable assignment.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 33 / 1

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SLIDE 96

Uniquely H-Saturated Graphs Computational Method

Symmetries of the System

The constraints

  • There is no r-clique in G.
  • Every non-edge e of G has exactly one r-clique in G + e.

are independent of vertex labels.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 34 / 1

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SLIDE 97

Uniquely H-Saturated Graphs Computational Method

Symmetries of the System

The constraints

  • There is no r-clique in G.
  • Every non-edge e of G has exactly one r-clique in G + e.

are independent of vertex labels. Automorphisms of the tricolored graph define orbits on variables xi,j.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 34 / 1

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SLIDE 98

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

Orbital branching reduces the number of isomorphic duplicates.

(Ostrowski, Linderoth, Rossi, Smriglio, 2007)

Generalizes branch-and-bound strategy from Integer Programming.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 35 / 1

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SLIDE 99

Uniquely H-Saturated Graphs Orbital Branching

Branch-and-Bound

x is given

Variable xi,j is selected

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 36 / 1

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SLIDE 100

Uniquely H-Saturated Graphs Orbital Branching

Branch-and-Bound

x is given

Variable xi,j is selected xi,j = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 36 / 1

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SLIDE 101

Uniquely H-Saturated Graphs Orbital Branching

Branch-and-Bound

x is given

Variable xi,j is selected xi,j = 0 xi,j = 1

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 36 / 1

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SLIDE 102

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

x is given

Orbit O is selected

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 37 / 1

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SLIDE 103

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

x is given

Orbit O is selected

xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 37 / 1

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SLIDE 104

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

x is given

Orbit O is selected

in orbit xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 37 / 1

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SLIDE 105

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

x is given

Orbit O is selected

in orbit xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 37 / 1

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SLIDE 106

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching

x is given

Orbit O is selected

in orbit xi,j = 1 for all {i, j} ∈ O xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 37 / 1

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SLIDE 107

Uniquely H-Saturated Graphs Orbital Branching

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch Orbital Branching Problem

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 38 / 1

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SLIDE 108

Many nodes. Small computation per node.

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SLIDE 109

Fewer nodes. More computation per node.

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SLIDE 110

Uniquely H-Saturated Graphs Orbital Branching

Kr-Completions

For every non-edge we add, we add a Kr-completion: xi,j = 0 if and only if there exists a set S ⊂ [n], |S| = r − 2, so that xi,a = xj,a = xa,b = 1 for all a, b ∈ S.

S S S

r = 4 r = 5 r = 6

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 40 / 1

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SLIDE 111

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

x is given Orbit O is selected

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 41 / 1

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SLIDE 112

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

x is given Orbit O is selected

in orbit xi,j = 1 for all {i, j} ∈ O xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 41 / 1

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SLIDE 113

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

x is given Orbit O is selected

in orbit xi,j = 1 for all {i, j} ∈ O xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0 xi1,a=1 xj1,a=1 xa,b=1

(a, b in S1)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S2)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S3)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S4)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S5)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in St )

S1 S2 S3 S4 S5 St

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 41 / 1

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SLIDE 114

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

x is given Orbit O is selected

in orbit in orbit in orbit in orbit xi,j = 1 for all {i, j} ∈ O xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0 xi1,a=1 xj1,a=1 xa,b=1

(a, b in S1)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S2)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S3)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S4)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S5)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in St )

S1 S2 S3 S4 S5 St

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 41 / 1

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SLIDE 115

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

x is given Orbit O is selected

in orbit in orbit in orbit in orbit xi,j = 1 for all {i, j} ∈ O xi1,j1 = 0 xi2,j2 = 0 xi3,j3 = 0 xik,jk = 0 xi1,a=1 xj1,a=1 xa,b=1

(a, b in S1)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S2)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S3)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S4)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in S5)

xi1,a=1 xj1,a=1 xa,b=1

(a, b in St )

S1 S2 S3 S4 S5 St

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 41 / 1

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SLIDE 116

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 117

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Base Case

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 118

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Non-edge?

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 119

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 120

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 121

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 122

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Edge?

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 123

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions in orbit −

→

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 124

Uniquely H-Saturated Graphs Orbital Branching

Orbital Branching with Kr-Completions

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 42 / 1

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SLIDE 125

Uniquely H-Saturated Graphs Orbital Branching

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch Orbital Branching Custom Augmentations Problem

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 43 / 1

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SLIDE 126

Uniquely H-Saturated Graphs Orbital Branching

Exhaustive Search Times

n r = 4 r = 5 r = 6 r = 7 r = 8 10 0.10 s 0.37 s 0.13 s 0.01 s 0.01 s 11 0.68 s 5.25 s 1.91 s 0.28 s 0.09 s 12 4.58 s 1.60 m 25.39 s 1.97 s 1.12 s 13 34.66 s 34.54 m 6.53 m 59.94 s 20.03 s 14 4.93 m 10.39 h 5.13 h 20.66 m 2.71 m 15 40.59 m 23.49 d 10.08 d 12.28 h 1.22 h 16 6.34 h 1.58 y 1.74 y 34.53 d 1.88 d 17 3.44 d 8.76 y 115.69 d 18 53.01 d 19 2.01 y 20 45.11 y Total CPU times using Open Science Grid.

(≈ 8.83 × 1018 connected graphs of order 20)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 44 / 1

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SLIDE 127

Uniquely H-Saturated Graphs Orbital Branching

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch Orbital Branching Custom Augmentations Problem Computation Time

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 45 / 1

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SLIDE 128

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

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SLIDE 129

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs

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SLIDE 130

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs Cycle complements

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SLIDE 131

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs Cycle complements Old examples

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SLIDE 132

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs Cycle complements Old examples

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SLIDE 133

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs Cycle complements Old examples New examples

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SLIDE 134

← − clique size − → ← − vertices − →

n \ r 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Empty graphs Cycle complements Old examples New examples

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SLIDE 135

Uniquely H-Saturated Graphs 4-Primitive Graphs

4-Primitive Graphs

n = 13

G(A)

13

Paley(13)

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 47 / 1

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SLIDE 136

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 137

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 138

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 139

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 140

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 141

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 142

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 143

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 144

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 145

Uniquely H-Saturated Graphs 5-Primitive Graph

5-Primitive Graph

n = 16 : G(A)

16

Not all r-primitive graphs are regular!

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 48 / 1

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SLIDE 146

Uniquely H-Saturated Graphs 5-Primitive Graph

7-Primitive Graph

n = 17 : G(A)

17

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 49 / 1

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SLIDE 147

Uniquely H-Saturated Graphs 5-Primitive Graph

7-Primitive Graph

n = 17 : G(A)

17

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 49 / 1

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SLIDE 148

Uniquely H-Saturated Graphs Cayley Complements

The Cayley complement C(Zn, S) has vertex set {0, 1, . . . , n − 1} and an edge ij if and only if |i − j| (mod n) / ∈ S.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 50 / 1

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SLIDE 149

Uniquely H-Saturated Graphs Cayley Complements

The Cayley complement C(Zn, S) has vertex set {0, 1, . . . , n − 1} and an edge ij if and only if |i − j| (mod n) / ∈ S. For r ≥ 1, C(Z2r−1, {1}) ∼ = C2r−1 is r-primitive. C5 C7 C9

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 50 / 1

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SLIDE 150

Uniquely H-Saturated Graphs Cayley Complements

Searching for r-Primitive Cayley Complements

To search for Cayley complements C(Zn, S) with |S| = g:

  • 1. Select a generator set S = {a1 = 1 < a2 < a3 < · · · < ag} ⊆ Z.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 51 / 1

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SLIDE 151

Uniquely H-Saturated Graphs Cayley Complements

Searching for r-Primitive Cayley Complements

To search for Cayley complements C(Zn, S) with |S| = g:

  • 1. Select a generator set S = {a1 = 1 < a2 < a3 < · · · < ag} ⊆ Z.
  • 2. Select an integer n > 2ag.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 51 / 1

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SLIDE 152

Uniquely H-Saturated Graphs Cayley Complements

Searching for r-Primitive Cayley Complements

To search for Cayley complements C(Zn, S) with |S| = g:

  • 1. Select a generator set S = {a1 = 1 < a2 < a3 < · · · < ag} ⊆ Z.
  • 2. Select an integer n > 2ag.
  • 3. Compute r = ω(C(Zn, S)) + 1.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 51 / 1

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SLIDE 153

Uniquely H-Saturated Graphs Cayley Complements

Searching for r-Primitive Cayley Complements

To search for Cayley complements C(Zn, S) with |S| = g:

  • 1. Select a generator set S = {a1 = 1 < a2 < a3 < · · · < ag} ⊆ Z.
  • 2. Select an integer n > 2ag.
  • 3. Compute r = ω(C(Zn, S)) + 1.
  • 4. Check if C(Zn, S) + {0, ai} has a unique r-clique for all ai ∈ S.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 51 / 1

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SLIDE 154

Uniquely H-Saturated Graphs Cayley Complements

Searching for r-Primitive Cayley Complements

To search for Cayley complements C(Zn, S) with |S| = g:

  • 1. Select a generator set S = {a1 = 1 < a2 < a3 < · · · < ag} ⊆ Z.
  • 2. Select an integer n > 2ag.
  • 3. Compute r = ω(C(Zn, S)) + 1.
  • 4. Check if C(Zn, S) + {0, ai} has a unique r-clique for all ai ∈ S.

Used Niskanen and ¨ Osterg˚ ard’s cliquer software to compute ω(G).

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 51 / 1

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SLIDE 155

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

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 52 / 1

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SLIDE 156

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.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 53 / 1

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SLIDE 157

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.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 53 / 1

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SLIDE 158

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.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 53 / 1

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SLIDE 159

Uniquely H-Saturated Graphs Cayley Complements

Computational Combinatorics

Computational Combinatorics Algorithms High Performance Computing Pure Combinatorics TreeSearch Orbital Branching Custom Augmentations Problem Theorems Computation Time

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 54 / 1

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SLIDE 160

Uniquely H-Saturated Graphs Cayley Complements

What Next?

Technique-specific

  • 1. Orbital Branching: Formalize custom augmentations for arbitrary

constraint systems. Apply to problems like strongly regular graphs.

  • 2. Discharging: Automate process so computer can discover and

write proofs.

  • 3. More Techniques: Find, Adapt, or Develop.

Stephen Hartke (UNL) Uniquely Kr -Saturated Graphs 55 / 1

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SLIDE 161

Computational Combinatorics and the search for Uniquely Kr-Saturated Graphs Stephen G. Hartke

University of Nebraska–Lincoln, USA hartke@math.unl.edu http://www.math.unl.edu/∼shartke2/

Joint work with Stephen G. Hartke, appeared in Electronic Journal of Combinatorics, 2012.