5-5-2016 Acknowledgements Dr. John Caughman, Chair Committee - - PowerPoint PPT Presentation

James Mahoney Dissertation Adviser: Dr. John Caughman Portland State University 5-5-2016

Acknowledgements

 Dr. John Caughman, Chair  Committee members:

 Dr. Nirupama Bulusu  Dr. Derek Garton  Dr. Paul Latiolais  Dr. Joyce O’Halloran

 PSU Math Department, Enneking Fellowship Comm.  Friends and Family

Overview

1.

Introduction

2.

Background

3.

The Tree Graph Function and Parameters

4.

Properties of Tree Graphs

5.

Trees and Matchings in Complete Graphs

Introduction – Background – T(G) function – Properties - Matchings

Introduction

 Tree graphs first introduced by Cummins in 1966  ~20 major papers published since then  No one has systematically constructed them before  My two years of research builds on data from dozens of

examples

Introduction – Background – T(G) function – Properties - Matchings

Overview

1.

Introduction

2.

Background

3.

The Tree Graph Function and Parameters

4.

Properties of Tree Graphs

5.

Trees and Matchings in Complete Graphs

Introduction – Background – T(G) function – Properties - Matchings

Graphs and Spanning Trees

 Graphs have vertices and edges  Trees are connected graphs with no cycles  Spanning trees have the same vertices as the original graph  If a graph has 𝑜 vertices then a spanning tree will have

𝑜 − 1 edges

Introduction – Background – T(G) function – Properties - Matchings

Tree Graphs

 Let 𝐻 be a graph. The tree graph of 𝐻, 𝑈(𝐻), has

vertices which are the spanning trees of 𝐻, where two vertices are adjacent if and only if you can change from

• ne to the other by moving exactly one edge.

Introduction – Background – T(G) function – Properties - Matchings

Example: 𝐷4

Introduction – Background – T(G) function – Properties - Matchings

Example: 𝐷4

Introduction – Background – T(G) function – Properties - Matchings

Example: 𝐷4

𝑈 𝐷4 = 𝐿4

Introduction – Background – T(G) function – Properties - Matchings

Overview

1.

Introduction

2.

Background

3.

The Tree Graph Function and Parameters

4.

Properties of Tree Graphs

5.

Trees and Matchings in Complete Graphs

Introduction – Background – T(G) function – Properties - Matchings

Tree Graph Function & Parameters

 Thm (Liu, 1992):

𝜆 𝑈 𝐻 = 𝜆′ 𝑈 𝐻 = 𝜀(𝑈 𝐻 )

 Tree graphs are as connected as possible -

hard to break apart by removing vertices or edges

Introduction – Background – T(G) function – Properties - Matchings

Graphs with Cut Vertices

 Let 𝐻 and 𝐼 be graphs and let 𝐻 ⊙ 𝐼 be a graph that

joins a vertex in 𝐻 with a vertex in 𝐼.

 Thm: 𝑈 𝐻 ⊙ 𝐼 ≅ 𝑈(𝐻)□𝑈(𝐼).

 Tree graphs of joined graphs are the product of the tree

graphs of the pieces

Introduction – Background – T(G) function – Properties - Matchings

Realizing Tree Graphs

 Given 𝑈(𝐻), can we find a graph 𝐼 such that

𝑈 𝐼 ≅ 𝑈(𝐻)?

 What is the pre-image of a tree graph?

Introduction – Background – T(G) function – Properties - Matchings Where do I come from?

Isomorphic Tree Graphs

 These pairs of graphs are not isomorphic, but their

tree graphs are.

 The starting graphs are isoparic: they have the same

number of vertices and same number of edges but are not isomorphic.

Introduction – Background – T(G) function – Properties - Matchings

Isomorphic Tree Graphs

 These pairs of graphs are not isomorphic, but their

tree graphs are.

 The starting graphs are isoparic: they have the same

number of vertices and same number of edges but are not isomorphic.

Introduction – Background – T(G) function – Properties - Matchings

Realizing Tree Graphs

 These two graphs are isoparic and their tree graphs are

isoparic (both have 64 vertices and 368 edges).

Introduction – Background – T(G) function – Properties - Matchings

Isomorphic Tree Graphs

 Is it ever the case that 𝐻 ≇ 𝐼 but 𝑈 𝐻 ≅ 𝑈(𝐼)?  Thm: If 𝐻 is 3-connected and planar, 𝑈 𝐻 ≅ 𝑈(𝐻∗).

Planar duals give isomorphic tree graphs.

Introduction – Background – T(G) function – Properties - Matchings

Tree Graph Function

Tree Graphs Starting Graphs Isoparic Isomorphic Neither Isoparic Isomorphic Never Always Never Neither ? Non planar duals? Default Introduction – Background – T(G) function – Properties - Matchings

Overview

1.

Introduction

2.

Background

3.

The Tree Graph Function and Parameters

4.

Properties of Tree Graphs

5.

Trees and Matchings in Complete Graphs

Introduction – Background – T(G) function – Properties - Matchings

Properties of Tree Graphs

 Thm (Cummins, 1966):

𝑈(𝐻) is hamiltonian for any graph 𝐻

 There is a cycle through all of the vertices

Introduction – Background – T(G) function – Properties - Matchings

Symmetry of Tree Graphs

 An automorphism of a graph 𝐻 is a permutation of the

vertices that respects adjacency. The set of all automorphisms of 𝐻 forms a group under composition, 𝐵𝑣𝑢(𝐻).

 The glory of a graph 𝐻, 𝑕(𝐻), is the size of its

automorphism group. 𝑕 𝐻 = |𝐵𝑣𝑢 𝐻 |.

 𝑕(𝐻) has been large for most of the small graphs studied

so far.

Introduction – Background – T(G) function – Properties - Matchings

𝐵𝑣𝑢(𝑈 𝐻 )

 Thm: 𝐵𝑣𝑢(𝐻) is a subgroup of 𝐵𝑣𝑢(𝑈 𝐻 ).

 The symmetries of the input are mirrored in the symmetries

• f the output.

 Example: 𝐵𝑣𝑢 𝐿4 − 𝑓 ≅ 𝑊

4 while 𝐵𝑣𝑢 𝑈 𝐿4 − 𝑓

≅ 𝐸8, the symmetries of the square.

Introduction – Background – T(G) function – Properties - Matchings

Summary of Proof

 Every graph automorphism 𝜏 of 𝐻 induces a tree graph

automorphism 𝜚𝜏 of 𝑈(𝐻)

 If 𝜚𝜏 fixes all vertices of 𝑈(𝐻), then 𝜏 fixes all cycle edges

• f 𝐻

 In a 2-connected graph, all edges are cycle edges  If all edges of 𝐻 are fixed by 𝜏, all vertices are fixed also  Therefore map that takes 𝜏 to 𝜚𝜏 is an injective

homomorphism

Introduction – Background – T(G) function – Properties - Matchings 𝐵𝑣𝑢(𝐻) 𝐵𝑣𝑢(𝑈 𝐻 )

Automorphism Size Examples

Graph 𝐻 g 𝑼 𝑯 𝒉 𝑯 Notes 8 4 𝐸8 and 𝑊

4

𝐿3,2 48 12 𝑇4 × 𝑇2 and 𝑇3 × 𝑇2 𝐿5 120 120 𝑇5 and 𝑇5 28800 4 ? and 𝑊

4

288 3 ? and ℤ3 12 1 𝐸12 and trivial 𝐷4 24 8 𝑇4 and 𝐸8 Introduction – Background – T(G) function – Properties - Matchings

Planarity

 Thm: The tree graphs of the diamond and the butterfly

are nonplanar. (Contain 𝐿5 and 𝐿3,3 minors, respectively.)

 Thm: 𝑈(𝐻) is nonplanar unless 𝐻 ≅ 𝐷3, 𝐷4.

 Cannot draw them flat without lines crossing.

Diamond Butterfly Introduction – Background – T(G) function – Properties - Matchings 𝑈 𝐼 ≤ 𝑈 𝐻 𝐼 ⊑ 𝐻

Decomposition

 Thm: The edges of 𝑈(𝐻) can be decomposed into

cliques of size at least three such that each vertex is in exactly 𝑛 − 𝑜 + 1 cliques.

 Can break apart graph into pieces that are completely

connected, where each vertex is in same number of pieces.

 Can be used to predict number of edges in 𝑈(𝐻).

Introduction – Background – T(G) function – Properties - Matchings

Decomposition

𝑛 = 5 𝑜 = 4 𝑛 − 𝑜 + 1 = 2 Introduction – Background – T(G) function – Properties - Matchings

Additional Families

 Let 𝑄𝑜,𝑙 be the graph where two vertices are joined by

𝑜 disjoint paths of edge length 𝑙.

 Thm: 𝑈(𝑄𝑜,𝑙) is (𝑜 − 1)(2𝑙 − 1)-regular.  Conj: 𝑈(𝑄𝑜,𝑙) is integral (with easily-understood

eigenvalues) and vertex transitive.

 𝑈(𝑄𝑜,𝑙) could be a new infinite family (with two

parameters) of regular integral graphs.

 These are really nice graphs

𝑄3,4

Introduction – Background – T(G) function – Properties - Matchings 𝑈(𝑄3,2)

Overview

1.

Introduction

2.

Background

3.

The Tree Graph Function and Parameters

4.

Properties of Tree Graphs

5.

Trees and Matchings in Complete Graphs

Introduction – Background – T(G) function – Properties - Matchings

Doyle Graph Coxeter Graph

• Def. A perfect matching is a set of disjoint edges that

covers all of the vertices in a graph.

Introduction – Background – T(G) function – Properties - Matchings

Coloring the edges of a graph

A coloring is an assignment of colors (numbers) to the edges of a graph A proper coloring has distinct colors at each vertex. Notice that the color classes for a proper coloring must be disjoint sets of edges (= matchings!)

1 5 6 4 5 2 3 2

Introduction – Background – T(G) function – Properties - Matchings

1-factorizations of 𝐿2𝑜

 Lots of not-so-nice ones…

In fact, of the 396 different rainbow colorings of 𝐿10, most look ‘random’

 Some very nice ones…

The most commonly known rainbow coloring of 𝐿2𝑜 is called 𝐻𝐿2𝑜

Introduction – Background – T(G) function – Properties - Matchings

The 𝐻𝐿2𝑜 1-factorization

Introduction – Background – T(G) function – Properties - Matchings

Orthogonal spanning trees

For any 1-factorization of 𝐿2𝑜, an orthogonal spanning tree has no 2 edges of the same color! (2𝑜 − 1 different colors)

Introduction – Background – T(G) function – Properties - Matchings

Brualdi-Hollingsworth Theorem

• Thm. (1996) Any 1-factorization of 𝐿2𝑜 has at least 2

disjoint orthogonal spanning trees.

Introduction – Background – T(G) function – Properties - Matchings

Brualdi-Hollingsworth Conjecture

• Conj. (1996) Any 1-factorization of 𝐿2𝑜 has a full set of 𝑜

disjoint orthogonal spanning trees!

Introduction – Background – T(G) function – Properties - Matchings

A first step

• Thm. (Krussel, Marshall, and Verall, 2000)

Any 1-factorization of 𝐿2𝑜, has at least 3 disjoint orthogonal spanning trees!

Introduction – Background – T(G) function – Properties - Matchings

• Thm. (KMV, 2000) If 2𝑜 − 1 is a prime of the form

8𝑛 + 7 then 𝐻𝐿2𝑜 has a full set of 𝑜 disjoint orthogonal spanning trees.

Another step

Introduction – Background – T(G) function – Properties - Matchings

 Since 𝐻𝐿2𝑜 is so nice, the symmetry should help us

build nice trees, too.

 Specifically, the colorings rotate around a single vertex.

So perhaps the trees should, too.

An idea to build upon

Introduction – Background – T(G) function – Properties - Matchings

WK28

Rotational 1-factorizations

• Def. In a rotational 1-factorization, each 𝑁𝑗, can be
• btained from 𝑁1 by rotation.

Introduction – Background – T(G) function – Properties - Matchings

Rotational spanning trees

• Def. In a rotational set of spanning trees all (but one) of

the trees 𝑈𝑗, can be obtained from 𝑈1 by rotation.

Introduction – Background – T(G) function – Properties - Matchings

Proof of concept

• Thm. (Caughman, Krussel) For every 𝑜, 𝐻𝐿2𝑜 has a full

rotational set of 𝑜 disjoint orthogonal spanning trees.

Introduction – Background – T(G) function – Properties - Matchings

New 1-Factorization

 Called the half family, 𝐼𝐿2𝑜

Introduction – Background – T(G) function – Properties - Matchings

Proposed Extension

• Conj. Every rotational 1-factorization of 𝐿2𝑜 has a full

rotational set of orthogonal spanning trees.

Introduction – Background – T(G) function – Properties - Matchings

Thanks!

 Any questions?

𝑈(𝑄

4,2)