combinatorial type problems for triangulation graphs

This may be used to solve the combinatorial type problem in a variety of situations, most significantly in graphs with unbounded degree.. Roughly speaking, the extremal metric finds the

THE FLORIDA STATE UNIVERSITYCOLLEGE OF ARTS AND SCIENCES

COMBINATORIAL TYPE PROBLEMS FOR TRIANGULATION

GRAPHS

ByWILLIAM E WOOD

A Dissertation submitted to theDepartment of Mathematics

in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Degree Awarded:

Summer Semester, 2006

UMI Number: 3232464

3232464 2006

Copyright 2006 by Wood, William E.

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by ProQuest Information and Learning Company

The members of the Committee approve the dissertation of William E Wooddefended on July 6, 2006.

Philip BowersProfessor Directing Dissertation

Lois HawkesOutside Committee Member

Steve BellenotCommittee Member

Eric KlassenCommittee Member

Craig NolderCommittee Member

Jack QuineCommittee Member

The Office of Graduate Studies has verified and approved the above named committee members

Dedicated to my parents, Bob and Sue Wood, and to my godmother, Dee Gelbach,for offering their love and support through everything I’ve done Even the very silly

things

I owe a lot to a great many people for getting me this far

Thanks first go to my committee, Steve Bellenot, Phil Bowers, Lois Hawkes, EricKlassen, Craig Nolder, and Jack Quine for enduring reading this document

I have dedicated this work to my parents, Bob and Sue Wood, and my Aunt Dee

I would be nowhere without them

Ken Stephenson has been like a second advisor to me Monica Hurdal taught

me lots in my early graduate career, and I thank her for allowing me to work asher research assistant Ara Basmajian, Soo Bong Chae, Peter Doyle, Zheng-Xu He,Karsten Henckell, David Mullins, Eirini Poimenidiou, and Jim Tanton were some earlyinfluences without whom I would never have gotten here I also had a terrific group

of faculty and fellow graduate students at Florida State University whose counseland enthusiasm I have found invaluable Thanks also to David Dickerson for hiscontribution to this work

I also thank the many baristas at assorted Tallahassee coffee shops for transformingportions of my meager income into the caffeinated beverages that fueled this thesis.Finally, I owe a great debt to my advisor, Phil Bowers Phil has been a mentor

to me since long before I ever became a graduate student I am greatly honored tohave him as an advisor and as a friend

TYPESETTING NOTES

This thesis was written in LATEX 2εusing the excellent WinEdt 5 editor The circlepacking pictures were generated by Ken Stephenson’s CirclePack program Most ofthe remaining diagrams were drawn in XFig 3.2, with some drawn in Maple 9.5,OpenOffice 1.1.4, and Mayura 4.3

TABLE OF CONTENTS

List of Figures viii

Abstract ix

1 INTRODUCTION 1

2 DISCRETE CONFORMAL TYPE 4

2.1 Classical conformal type 4

2.2 From surfaces to graphs 6

2.3 Vertex and edge extremal length 8

2.4 Random walks and electric networks 10

2.5 Circle packings 12

2.6 Equilateral type 19

2.7 Equivalences 19

2.8 The type problem 21

3 BOUNDED REFINEMENTS 24

3.1 Combinatorial extremal length 24

3.2 Extremal metrics 26

3.3 Shadow paths 28

3.4 Extremal length of graphs 30

3.5 Electrical versus VEL type 32

3.6 Bounded refinements 35

3.7 Refinement and edge extremal length 39

3.8 Outdegree of planar graphs 41

3.9 Refinement and vertex extremal length 44

3.10 Refinement and type 46

3.11 Beyond triangulations 53

4 TYPE INVARIANTS 57

4.1 Parabolic vertex growth 57

4.2 Hyperbolic trees 59

4.3 Explicit parabolic extremal metrics 61

4.4 Growth versus asymmetry: displacing growth 62

4.5 Growth versus asymmetry: trapping growth 64

4.6 Slow growth in hyperbolic graphs 66

4.7 k-fuzz 69

4.8 Dual graphs 73

4.9 Outer spheres 74

5 EQUILATERAL SURFACES 80

5.1 Quasiconformal mappings 80

5.2 Equilateral surfaces 83

5.3 Bounded degree convergence 84

5.4 Unbounded degree convergence 86

6 CONCLUSION 90

APPENDIX: A CONFORMAL EXTREMAL METRIC ON THE PLANE 92

REFERENCES 95

BIOGRAPHICAL SKETCH 98

LIST OF FIGURES

2.1 Conducting surfaces and the Riemann map 12

2.2 The constant degree five, six, and seven packings 14

2.3 Circle packings approximate the Riemann map 18

2.4 Conductance for the tailored random walk 21

2.5 Packing with alternating generations of degree 6 and 7 22

3.1 The diamond configuration Dn 33

3.2 Resistors in series and parallel 34

3.3 Construction of a VEL-parabolic, EEL-hyperbolic electric network 36

3.4 Incident-adjacent edges 37

3.5 Barycentric and hexagonal refinement 38

3.6 Shadow paths in hex refinement 46

3.7 Adding a binary tree with an unbounded refinement 47

3.8 Zig-zag refinement 49

4.1 Circuit reductions for a binary tree 60

4.2 Sector of the constant degree six triangulation 61

4.3 A parabolic tree with exponential growth 63

4.4 Trapping growth inside diamonds 65

4.5 Quadratic growth in a hyperbolic triangulation 68

4.6 Parabolic graphs with hyperbolic k-fuzz 72

4.7 The graph G# is a refinement of both G and G∗ 73

4.8 Geodesic vertex paths in S(n) may not extend 75

4.9 A vertex in SO(n) has two neighbors in SO(n) 76

4.10 A vertex in SO(n) cannot have three neighbors in SO(n) 77

4.11 SO(n) is connected 78

4.12 The outer sphere skeleton GO 79

A.1 log log z as a conformal mapping 94

The main result in this thesis bounds the combinatorial modulus of a ring in atriangulation graph in terms of the modulus of a related ring The bounds dependonly on how the rings are related and not on the rings themselves This may be used

to solve the combinatorial type problem in a variety of situations, most significantly

in graphs with unbounded degree Other results regarding the type problem arepresented along with several examples illustrating the limits of the results

CHAPTER 1 INTRODUCTION

There is a dichotomy in mathematics between continuous and discrete objects.Continuous objects lend themselves to study by analysts and geometers, whereasdiscrete objects are the domain of algebraists and combinatorists We see the value

of bridging these two worlds in our first calculus classes when we see how to use acollection of well-chosen rectangles to approximate the area under a curve and thenlearn how to define area as a limit of a discrete approximation

Manifold topologists have found great value in the discrete approach Bydescribing a topological space in terms of how to build it out of glued simplices,one can uniquely describe a space combinatorially This is what makes algebraictopology work, and it has worked quite well

Much modern topology has explored how the topology of a space determines itsgeometry as well But if a topological space can be described combinatorially, andthe topology prescribes the geometry, surely this geometry must somehow be builtinto the combinatorics This is where the field of discrete conformal geometry begins.Considerable progress has been made toward carrying well-understood ideas fromclassical geometry and analysis and bringing them to the discrete world The goal

of this thesis is to address a discrete version of a classical question in conformalgeometry: the type problem

The Riemann Mapping Theorem tells us that a non-compact simply connectedRiemann surface is conformally equivalent to either the disk (hyperbolic) or the plane(parabolic), but it is often difficult to determine for a particular surface which one.This determination is the classical type problem

To get to the discrete version, we triangulate our surface The 1-skeleton of thistriangulation, called a triangulation graph, is our fundamental discrete object We

ask if there is some sense in which the graph naturally wants to be drawn in the disk

or the plane, and this should have something to do with the surface we triangulated

We will see that many theorems, techniques, and much of our intuition from Riemannsurfaces can be interpreted discretely

Our broad goal is to understand the nature of the type dichotomy For example, ifone has a graph of known type, to what extent could one modify it without changingthe type? Our main theorem gives a fairly precise answer to this question We alsowill present several examples illustrating exactly how far is too far

Many of the results of discrete conformal geometry use geometric methods,specifically those of quasiconformal mappings The problem is that these methodsusually require the graphs to have bounded degree Many of these theorems havebeen generalized to unbounded degree graphs, although the proofs get considerablymore complicated

Our approach will be more combinatorial We avoid the bounded degree condition

by replacing it with a weaker notion of degree, one that turns out to be shared by allplanar graphs This allows us to prove a very general result without the backbreakingmachinery We then survey how to use this and other techniques to solve the typeproblem for certain classes of graphs We also examine another form of combinatorialtype introduced by Bowers and Stephenson that offers a direct connection betweenthe combinatorics and the geometry

Chapter 2 surveys the field and introduces the various forms of conformal typeand how they may be applied discretely We also fix a lot of our working definitionsand notation The goal of Chapter 3 is to prove our featured theorem, along the waydeveloping a collection of useful techniques and examples In Chapter 4, we turnour attention to solving the type problem in specific cases and examining the impact

of various properties of a graph on its type The tone of our discussion changes

in Chapter 5 where we cover Bowers and Stephenson’s construction of a piecewiseflat Riemann surface from a triangulation graph We take a geometric approach toexamine the convergence behavior of these surfaces under refinement Chapter 6 isthe conclusion, where we explore where to take some of the ideas we develop herein

Our story borrows a lot from other fields, including complex analysis, geometry,graph theory, and even physics We try to survey the basics of the material used, butalways offer references when we are unable to provide full details Ken Stephenson’sbook on circle packings [Ste05] would be a valuable companion reference for thisthesis The general rule is that, unless stated otherwise, all proofs presented areoriginal Some are new proofs or minor generalizations of established results, orproofs of lemmas that are commonly accepted but seemingly undocumented In anycase, all appropriate references are introduced with the theorems.

CHAPTER 2 DISCRETE CONFORMAL TYPE

We overview the various forms of type in this chapter Our goal is to get a feel forthe objects of concern and to motivate the forthcoming results Some of the detailsare bypassed for material that will be developed later, will not play a role in ourproofs, or for which there are established references (which we provide) We also get

a few technical definitions and notations out of the way

2.1 Classical conformal type

We begin with a review of classical conformal type, presenting here only what isneeded to appreciate the forthcoming analogy to the discrete setting A somewhatmore thorough overview is offered in Chapter 5, or see the classical texts (e.g.,[LV70],[Vai71]) for more details

A metric on a domain S ⊂ C is a Borel measurable function ρ : S → [0, ∞] witharea defined by area(ρ) =RR

Sρ2 dA, where dA denotes Lebesgue measure on S Werequire area(ρ) > 0 and a metric is admissible if area(ρ) is finite Denote by M(S)the set of all admissible metrics on S

A path in S is a continuous mapping of the interval [0, 1] into S, with a path toinfinity being a path containing the point at infinity We further assume that all pathsare locally rectifiable For a metric ρ, a path γ in S has ρ-length Lρ(γ) =R

γρ dz, theintegral again taken with respect to Lebesgue measure For any collection Γ of paths

in S, define the extremal length

EL(Γ) = sup

ρ∈M(S)

infγ∈ΓLρ(γ)2area(ρ) .The reciprocal of extremal length is the modulus, denoted Mod(Γ) It has the followingproperties

Theorem 2.1 The modulus defines an outer measure on the set of all curve families

In the special case that a ring domain R is a round annulus bounded by twoconcentric circles with radii r1 and r2, r1 > r2, we have Mod(R) = logr1

r 2 Moregenerally, we note that for any ring domain R there is a conformal mapping ffrom R onto a round annulus with radii unique up to scaling It turns out thatMod(R) = Mod(f (R)) In particular, the modulus is a conformal invariant It is easy

to see this from the definition by performing a change a variables in the integrals.This setup also works for quadrilaterals instead of ring domains A quadrilateral

Q is a Jordan domain in C with four distinguished boundary points separating thesides of the quadrilateral We define the modulus of the quadrilateral to be themodulus of the family of curves in Q connecting one of the pairs of opposite sides.The modulus again finds the conformal mapping of Q onto a euclidean rectangle, andthe modulus is the aspect ratio of this rectangle (Switching the choice of side pairsreciprocates the modulus.) Quadrilaterals relate to ring domains via the exponentialmap, and the extremal metric is realized by the derivative of a conformal mapping ofthe quadrilateral onto a euclidean rectangle mapping sides to corresponding sides

Fix a point s ∈ S and let A be the set of all ring domains separating s frominfinity (i.e., s is in the bounded component of C \ R for R ∈ A) We define

EL(S) = sup

R∈AEL(R)

We say S is parabolic if EL(S) = ∞ and hyperbolic if EL(S) < ∞ This classification isthe conformal type and is the same classification determined by the famous RiemannMapping Theorem

Theorem 2.2 (Riemann Mapping Theorem) Any open simply connected planar main may be mapped onto either C or the unit disk D via a conformal homeomor-phism This homeomorphism is unique up to automorphisms of the image set.Theorem 2.3 If S is hyperbolic, then there is a conformal homeomorphism of S onto

do-D Otherwise, S is parabolic and conformally equivalent to the plane

In fact, the supremum in the definition of extremal length is realized by thederivative of a conformal map of onto C or D

Roughly speaking, the extremal metric finds the finite metric with the “shortestpath to infinity.” We may think of the extremal length interpretation of type ascapturing the fact that area is accrued much faster in hyperbolic geometry than ineuclidean geometry

Although we have discussed the theory in terms of domains in C, the definitions

in this section extend easily to non-compact simply connected Riemann surfaces S.Let f be a conformal homeomorphism of S into a region in C The modulus, extremallength, and conformal type are defined on S as the corresponding notions in f (S),and the theorems discussed still hold

2.2 From surfaces to graphs

In the combinatorial setting, our fundamental objects are no longer surfaces, butgraphs We make this connection in the next section, but first we establish somedefinitions

A graph G = (V, E) is a set V of vertices and a set E of unordered pairs of verticescalled edges An edge connecting two vertices v0 and v1 is denoted [v0, v1], and thevertices v0 and v1 are called the endpoints Note that the definition and notationdisallow multiple edges connecting two vertices (a graph permitting multiple edges iscalled a multigraph) A graph is finite or infinite as |V | is finite or infinite We assumeour graphs have no edges connecting a vertex to itself (loops) Two vertices v, w areadjacent, denoted v ∼ w, if [v, w] ∈ E Two graphs G = (V, E) and G0 = (V0, E0)are isomorphic, denoted G ∼= G0, if there is a bijection f : V → V0 such that forany v, w ∈ V we have v ∼ w if and only if f (v) ∼ f (w) Every vertex has a degreedeg(v0) = |{v ∈ V : [v, v0] ∈ E}|, the number of edges having v0 as an endpoint Weassume the degree is finite for each vertex (the graph is locally finite) The degree of

a graph is defined as supv∈V deg(v) The graph has bounded degree if the degree isfinite, unbounded degree otherwise A cycle is a finite connected graph whose verticesall have degree two

A vertex path in G is a finite or infinite sequence of vertices v0, v1, such that forall i ≥ 0, either vi ∼ vi+1 or vi = vi+1 Similarly, an edge path is a sequence of edges

of the form [v0, v1], [v1, v2], [v2, v3], , i.e consecutive edges laid end to end Notethat each vertex path has a corresponding edge path, and vice versa Our graphs areassumed to be connected, so that there is a vertex path connecting any two vertices in

V The combinatorial distance between vertices v and w, denoted dcomb(v, w), is thelength of shortest edge path from v to w Thus, dcomb(v, v) = 0 and dcomb(v, w) = 1

if and only if v ∼ w The n-sphere about a vertex is the set of vertices w such that

dcomb(v, w) = n

We generally think of the vertices as points and the edges as arcs connecting theendpoints A graph is planar if the graph can be embedded in the plane – that is,the vertex points and edge arcs may be positioned in the plane so that the verticesare located at distinct points and the edge arcs intersect only at shared endpoints

An embedding like this is a diagram for a graph For example, the diagram of a cycle

is a Jordan curve

Our interest in graphs is to mimic the behavior of surfaces with a combinatorialobject This is achieved via the triangulation graph, which is the 1-skeleton of a

locally finite tiling by triangles of a simply connected Riemann surface (possibly withboundary) A triangle with vertices or edges a, b, c is denoted 4(a, b, c) Of specificinterest are disk triangulation graphs, in which the Riemann surface in question isthe open unit disk Most of our discussions begin (but not end) with consideration

of these graphs because of their connection to circle packings, which we discuss inSection 2.5 We also study more general disk cell complexes whose 1-skeletons arecalled disk cell graphs These are obtained from locally finite tilings of the disk bygeneral finite polygons, not necessarily triangles The interiors of the polygons arethe faces or cells We frequently require a global bound on the number of sides of thepolygonal faces When a cell structure is present and relevant, we expand the graphnotation G = (V, E, F ) to include the set of polygonal faces F

Let G be a cell complex with disjoint subcomplexes A, B, C ⊂ G We say Cseparates A from B if A and B lie in different components of G \ C We say Cseparates A from infinity if A lies in an unbounded component of G \ C

A cell complex has an associated complex formed by exchanging the roles ofvertices and faces A cell complex G = (V, E, F ) has a dual graph G∗ = (V∗, E∗, F∗)where V∗ = F , F∗ = V , and [f1, f2] ∈ E∗ if and only if f1 and f2 share an edge e1,2

We associate e1,2 with its dual edge e∗

2.3 Vertex and edge extremal length

The use of triangulations to describe a surface combinatorially is familiar fromalgebraic topology We want to strengthen this link to include the geometry as well

as the topology Our first tool is a direct analog of the classical extremal lengthdescribed in Section 2.1 This material is of fundamental importance to our results,

so we defer a careful development to Chapter 3

A vertex metric on a graph G = (V, E) is a function m : V → [0, ∞] and the area

of m is defined as area(m) = P

v∈V m(v)2 The length of a vertex path in a metric

m is Lm(γ) =P

v∈γm(v) Note that a single vertex in V may appear multiple times

in γ; such vertices are counted with multiplicity We shall maintain this abuse ofnotation throughout

We may think of a vertex metric as assigning positive weights to the vertices

of G The metric plays the same role as did positive measurable functions in theclassical setting In this discrete setting, however, there is another way to set up thedefinitions: instead of assigning weights to the vertices, we could assign them to tothe edges of G

An edge metric on a graph G is a function m : E → [0, ∞], with area(m) =P

e∈Em(e)2 The edge length of an edge path γ is Lm(γ) =P

e∈γm(e) As before, ametric is admissible if it has finite area

Let ME(G) (MV(G)) denote the sets of admissible edge (vertex) metrics, and let

ΓE (resp., ΓV) denote a collection of edge (vertex) paths in G We define the vertexextremal length of ΓV, denoted VEL(ΓV), by

VEL(ΓV) = sup

m∈M V

infγ∈ΓV Lm(γ)2area(m)and the edge extremal length, denoted EEL(ΓE) by

EEL(ΓE) = sup

m∈M E

infγ∈ΓE Lm(γ)2area(m)

We reserve the notations EEL(G) and VEL(G) for the case where ΓE (ΓV) is the set

of edge (resp., vertex) paths to infinity, i.e paths that start at a fixed vertex of Gand are not contained in any finite subset of G These are the transient paths in Gbased at v0 (When we say the transient paths of G, we are implicitly assuming abase vertex has been chosen.) As in the classical setting, a graph G is VEL-parabolic

if and only if VEL(G) = ∞, and otherwise is VEL-hyperbolic EEL-parabolic andEEL-hyperbolic are defined analogously

It is worth noting that although triangulations behave as our liaison between theclassical and discrete intuition, the definitions of edge and vertex extremal length

do not require that G be the 1-skeleton of a triangulation We will explore thecombinatorial types of graphs that are not cell complexes, or even planar.

Edge extremal length was introduced by Duffin in [Duf62] Vertex extremal lengthwas introduced by Cannon in [Can94], though our definitions are more consistentwith He and Schramm’s treatment in [HS95] See [CFP97] for nice introduction

to Cannon’s approach We see later how these objects behave similarly to theirconformal counterparts For example, a monotonicity property (Theorem 2.1 (ii))holds

Recall from Section 2.1 that the extremal metric in classical extremal length wasattained for the derivative of the Riemann mapping onto the disk or the plane Wemay also assign meaning to the metric at which the vertex extremal length is realised.Let Q be combinatorial quadrilateral – a combinatorial disk with four distinguishedboundary vertices It is shown in [CFP94] and [Sch93] that there is a euclideanrectangle that can be filled with squares so that the squares correspond to verticesand two squares share boundary when their corresponding vertices are adjacent.The vertex extremal metric on Q returns the size of these squares in the tiling andthe vertex extremal length is the aspect ratio of the rectangle (The curve familyassociated to the extremal length is the set of vertex paths connecting one of thepairs of opposite sides Compare with Section 2.1.)

One obvious question is, does it matter if we use vertices or edges? That is, canthe VEL type of a graph be different from its EEL type? We will see that the answer

is, inconveniently, yes

2.4 Random walks and electric networks

Let v0 be a vertex in a graph G = (V, E) and consider a walker beginning at v0who moves every second to an adjacent vertex For v, w ∈ V , we establish a transitionprobability function p(v, w) which gives the probability that a walker at v moves to

w in the next step We require p(v, w) > 0 if and only if [v, w] ∈ E This reflectsthe requirement that the walker may only move to adjacent vertices and may notstand still, and further imposes that all adjacent vertices are always accessible to thewalker (i.e., the walk is reversible) We also have P

w∈V p(v, w) = 1 for every v ∈ V

We then ask: what is the probability that a random walker will eventually return to

v0? The random walk is called recurrent if the return probability is 1 Otherwise thewalk is transient

This setup has another interpretation Consider the edges of our graph G asresistive wires and connect a 1-volt battery at v0 grounded at the boundary of G(possibly infinity) We can think of the electrons in the current as random walkerswandering through the network The transition probabilities depend on the resistance

of the wires Specifically,

p(v, x) = P c(v, x)

w∼vc(v, w)where c(v, w) denotes the conductance (reciprocal of resistance) of the edge (resistor)[v, w] That is, the transition probability from v is an edge’s conductance taken as

a fraction of the total conductance of all edges connected to v Recurrence in thiselectric network formulation means no charge escapes to infinity, or the resistance toinfinity is infinite (or, equivalently, the conductance to infinity is zero)

We now consider the special case in which a random walker chooses his next stepcompletely at random, so p(v, w) = deg v1 This is known as a simple random walk.Electrically, this means c(v, w) ≡ 1, or all of the edge-wires are unit resistors Weuse this case to define a version of combinatorial type: a graph G is RW-parabolic if

a simple random walk on G is recurrent, and G is RW-hyperbolic if a simple randomwalk is transient This determines the RW-type of G, which is the same as its electricaltype

These, too, have classical analogs Imagine a surface that is a topological annulusand is made from a conductive material Connect the two poles of a battery to the twoboundary circles of the surface This induces a current across the surface There must

be curves of electric flow and curves of equipotential, and they must be orthogonal

to each other See Figure 2.1 These curves define a local coordinate system Bymapping the flow curves to straight lines and the equipotential curves to circles, weconstruct the conformal mapping of the surface to a right circular cylinder Electricnetworks are the discrete analog of this interpretation Similarly, random walks are

a discrete analog of Brownian motion on surfaces

Figure 2.1: Conducting surfaces and the Riemann map.

The title of this section is that of the book by Doyle and Snell [DS84] in whichthe the connections between random walks, electric networks, and discrete harmonicfunctions are elegantly explored See also [Bol98]

Among their main results is an electrical proof of P´olya’s Theorem

Theorem 2.4 (P´olya’s Theorem) A simple random walk on the integer (square)lattice in the plane is recurrent, whereas the simple random walk on the integer lattice

in Rn is transient for n > 2

Their method of proof is to use circuit reductions to find hyperbolic trees inside theselattices We will explore such graphs in depth in Chapter 4, but note for now that thebinary tree is hyperbolic (We will use this fact before we get to Chapter 4, but ourresults there include this fact.) A nice way to prove something hyperbolic is to find

a binary tree (or some other hyperbolic graph) inside it and conclude hyperbolicityfrom monotonicity See also [Bow98] for a geometric application of this method

2.5 Circle packings

In this section, we survey the theory of circle packings and discrete analytic tions Circle packings offer a direct connection between geometry and combinatorics,and we appeal to our circle packing intuition to understand the theorems of discreteconformal geometry even when the proofs avoid any mention of circles (as shall be thecase for our results) Most of this exposition is cribbed from Ken Stephenson’s treatise

func-on the subject [Ste05], to which the reader is referred for a thorough treatment

A circle packing consists of a triangulation graph G = (V, E), called the contactsgraph or complex, and a collection of circles Cj, vj ∈ V (G), with disjoint interiorsarranged so that Ci ∩ Cj 6= ∅ if and only if [vi, vj] ∈ E(G) These circles may

be in spherical, hyperbolic, or euclidean geometry We are imposing here a standingassumption that the packing be univalent by requiring circles to have disjoint interiors.(Otherwise we could allow a chain of circles to wrap multiple times around a circle,giving a branched packing.) The basic properties of these geometries are summarized

in Table 2.5 Some familiarity with this material is assumed of the reader [Bea83] and[Sti92] are good background sources We always assume that every boundary vertex

of G is adjacent to at least one interior vertex (in particular, there is at least oneinterior vertex) (Note that the complex of a circle packing is always a triangulation

We use the term “complex” to describe more general cell structures, but never for

a circle packing.) For a circle c in a packing, its flower is c along with the circlestangent to c As we add layers of circles about c, we define the nth generation ofcircles about c as the circles whose combinatorial distance to c is n (technically wemean the combinatorial distance in the contacts graph of the vertices corresponding

to the circles) That is, we are thinking of the packing as growing out from c and thegeneration number indicates how long it takes to get to a particular circle

A circle packing determines a function R : V → (0, ∞), called a packing label,which associates to each vertex vi the radius of the circle Ci in the appropriategeometry (In the hyperbolic case, we permit infinite hyperbolic radii for boundaryvertices Such vertices correspond to horocycles in the circle packing.)

Since the contacts graph is a triangulation, all interior circles meet in triples Wemay draw the geodesics connecting the centers of these circles, giving a picture of thecontacts graph with the edges as geodesics and the triangular faces as triangles in theappropriate geometry The union of these triangles is the carrier of the circle packing.When a circle packing’s carrier is a surface S, often the open disk or the plane, wesay the packing fills S Figure 2.2 shows packings typical of the three geometries.Suppose two circle packings P1 and P2 have the same contacts graph in thesame geometry A discrete analytic function f is a continuous orientation-preservingmapping taking the carrier of P1 onto the carrier of P2 so that the center of a circle

(a) The constant degree five (dodecahedron) packing on the sphere.

(b) The constant degree six (penny) packing in the plane.

(c) The constant degree seven packing in the disk.

Figure 2.2: The spherical degree five packing, the parabolic degree six packing, and thehyperbolic degree seven packing with underlying contacts graphs drawn respecting thecorresponding geometry

Table 2.1: Summary of surface geometries

sine law sinh asin α = sinh bsin β = sinh csin γ sin αa = sin βb = sin γc sin αsin a = sin βsin b = sin γsin c

cosh a cosh b a2 + b2 cos a cos b

− sinh a sinh b cos γ −2ab cos γ − sin a sin b cos γautomorphisms z 7→ λ1−¯z−ααz, z 7→ az + b, SO(3, R)

|λ| = 1, α ∈ D a, b ∈ C (taking S2 ⊂ R3)isometries z 7→ λ1−¯z−ααz, z 7→ λz + τ, z 7→ az+bcz+d,

|λ| = 1, α ∈ D λ, τ ∈ C, |λ| = 1 a, b, c, d ∈ C,

ad − bc = 1

in P1 maps to the center of its corresponding circle in P2, and the map extendsbarycentrically to edges and faces Usually one of the packings will be of the disk,plane, or sphere

It is through these maps that circle packings approximate classical complexanalysis Whereas an analytic map takes infinitesimal circles to infinitesimal circles, adiscrete analytic function maps actual circles to actual circles (roughly – this definitiondoes not actually demand f take a circle setwise onto its counterpart, but there is awell-defined correspondence) There are discrete versions of such classical theorems asthe argument principle, the maximum principle, Schwarz’ lemma, and many others.The function returning the ratio the the radii of corresponding circles behaves like a

discrete derivative If we allow tangent circles to wrap more than once around a circle,

we open the exploration of branched functions Our interest is in the discrete analog

of one of the greatest theorems of complex analysis, the Uniformization Theorem.Theorem 2.5 (Discrete uniformization) Let G be a triangulation of a Riemannsurface S Then there is a surface of constant curvature S0 homeomorphic to Sand a circle packing in the appropriate hyperbolic, euclidean, or spherical metric withcontacts graph G and filling S0 S0 is unique up to conformal equivalence and thepacking is unique up to conformal automorphisms of S0

Our surfaces are simply connected The non-uniqueness of the packings is the samenon-uniqueness in classical uniformization We usually normalize by requiring thatsome specified circle be centered at the origin and some other circle be centered on thereal axis The theorem was proved by Beardon and Stephenson in [BS90] (boundeddegree) and He and Schramm in [HS96] (unbounded degree)

We are particularly interested in circle packings that fill the open disk or the plane

A packing for a complex G filling one of these surfaces is a maximal packing for G.Every disk triangulation graph has a maximal packing Furthermore, a finite planartriangulation of a closed disk may be realized as a packing in D so that the boundarycircles are horocycles This is also called a maximal packing

The analogy between discrete and classical conformal geometry is not just retical

theo-Theorem 2.6 Let S and S0 be simply connected regions of the plane and let P1, P2,

be a sequence of circle packings in S with contacts graphs G1, G2, Suppose

P0

1, P0

2, are packings of S0 with the same contacts graphs G1, G2, and so thatthere is a sequence of positive numbers δn → 0 so that the radii of the circles in Pnare less than δn and each boundary circle of Pn and P0

n lies within a distance of δn ofthe boundary of S and S0, respectively Let fn be the discrete analytic function from

Pn to P0

n Suppose for some point p in the carrier of Pn, the images fn(p) lie in acompact subset of S0 Then there is a subsequence of the fn converging uniformly oncompact subsets of S to a conformal homeomorphism of S onto S0

In other words, circle packings actually approximate the Riemann map The process

is illustrated in Figure 2.3

The methods involved in proving these theorems are worth noting In their originalproof of the first version of Theorem 2.6, Rodin and Sullivan [RS87] proved the RingLemma, which essentially says that the radii of n circles encircling the unit circlecannot be smaller than some constant depending on n ([Aha97] shows the dependence

on n is exponential) This essentially says that the maps between correspondingtriangles in the carriers are not overly distorted as long as the degree of the contactsgraph is bounded By “not overly distorted,” we mean specifically quasiconformal Wewill return to this topic in Chapter 5, noting for now that quasiconformal mappingshave some very convenient convergence properties that drive the proof He andSchramm [HS96] removed the requirement that the graph have bounded degree, butthey had to develop a completely different approach to do so This is typical; many

of the geometric methods completely fail for unbounded degree graphs A centraltheme of this thesis is to weigh the influence of this condition

An important ingredient to Rodin and Sullivan’s proof is the Hexagonal PackingLemma

Lemma 2.7 (hexagonal packing lemma) There is a sequence sn → 0 satisfying thefollowing Let c be a circle in a euclidean circle packing P and suppose the first ngenerations of circles about c are combinatorially equivalent to n generations of theconstant degree six packing Then

1 −

radius(c 0 ) radius(c)

≤ sn for any circle c0 tangent to c.This basically says that if we drill deep into a packing with expanses of hexagonalcombinatorics, the circles there have nearly identical radii, i.e the packing resemblesthe penny packing there This is part of why we want to prove theorems aboutprocesses that add lots of degree six vertices to a graph Such processes createregions that look euclidean Another interpretation we use is that the quasiconformaldilitation of the discrete analytic mapping (i.e., how far from conformal it is) takingthe triangles of n generations of degree six combinatorics onto euclidean equilateraltriangles tends to one

(a) A region in the plane is approximated by the penny packing.

(b) The corresponding maximal packing.

Figure 2.3: A region in the plane is approximated by the constant degree six packing ofthe plane We then take the maximal packing in the disk with the same contacts graph

As we take finer approximations of the region, the corresponding discrete analytic functionconverges to the conformal Riemann mapping of the region to the disk (The region here

is bounded by the polar curve r = 1 + sin2(32θ) and the circles have radius 503 Thesechoices are purely aesthetic.)

The proof relies heavily on the rigidity of the penny packing That is, the onlypossible packing labels for the constant degree six complex that yield a univalentpacking filling the plane demand that all circles have the same radius See [RS87] or[Ste05] for a proof, or [Aha90],[Aha94] for an alternate approach and generalizations.Our interest is in the discrete notion of conformal type the theory of circle packingsprovides A disk triangulation graph is circle pack hyperbolic (CP-hyperbolic) if it isthe contacts graph of a circle packing of the unit disk Otherwise it is the contactsgraph for a packing of the plane and the graph is CP-parabolic The theory tells usthat these possibilities are exhaustive and mutually exclusive.

2.6 Equilateral type

Equilateral type was introduced by Bowers and Stephenson in [BS04] Let G be

a disk triangulation graph and define a surface |G|eq by assigning to each triangularface of G the geometry of a flat unit equilateral euclidean triangle We can use anappropriate power map in vertex neighborhoods to define a metric and conformalstructure making |G|eq into a Riemann surface Details of this construction will bediscussed in Section 5.2 This surface has a classical conformal type as described inSection 2.1, and we define the equilateral type of G to be the classical conformal type

of |G|eq

Equilateral type is defined as the classical type of a surface that is describedpurely combinatorially As such, equilateral type offers a very direct link between theclassical and discrete theories Indeed, the connection is very natural In the absence

of any imposed geometry, the combinatorial triangles of a graph behave as thoughthey are equilateral An exploration of this link is the content of Chapter 5

2.7 Equivalences

One might hope that the different versions of combinatorial type discussed in thischapter would all be the same for any graph, but this is not quite so The equivalencesare summarized as follows:

Theorem 2.8 Let G be a disk triangulation graph

1 The VEL type of G is the same as its CP type.

2 The RW type, electrical type, and EEL type of G are the same

3 If G has bounded degree, then all described forms of combinatorial type –electrical type, RW type, CP type, VEL type, EEL type, and EQ type – arethe same

4 There is a graph that is EEL-hyperbolic and VEL-parabolic

In other words, there is really only one notion of type for a graph of bounded degreeand two for unbounded degree It is not known where EQ-type fits for unboundeddegree graphs

Parts (1) and (4) are proved by He and Schramm in [HS95] We detail theconstruction of a VEL-parabolic and EEL-hyperbolic graph in Chapter 3 Theequivalence between electrical and RW-type as outlined in Section 2.4 is detailed

in [DS84], whereas the equivalence of EEL type and electrical type may be found in[Duf62] For another proof of bounded degree equivalence of CP and RW types, see[McC98] Bowers and Stephenson prove the bounded degree equivalence of EQ-type

to CP-type in [BS04] (see also [Ste05]), a result we discuss and expand in Chapter 5.Another connection is from Tomasz Dubejko [Dub97], who introduces the tailoredrandom walk This has the same setup as the simple random walk except that wenow allow the circle packing to bias the walker’s choice of direction Let R be aeuclidean packing label for a graph G = (V, E) (even if the packing is in the disk, wetake the euclidean rather than the hyperbolic radii of the circles) The label realizeseach triangle v0, v1, v2 in G as a euclidean triangle To each edge eij = [vi, vj] ∈ E,define its length L(eij) = R(vi) + R(vj) to be the euclidean length of the edge in thepacking Now consider the triangles v, vi, vj and w, vi, vj that eij bounds Let L∗(eij)

be the distance between the incenters of these triangles (the incenter is the center ofthe inscribed circle) and define C(e) = LL(e)∗(e) These are the conductances of randomwalk, so that a walker at v moves to an adjacent vertex v0 with probability

p(v, v0) = C([v, v

0])P

w∼vC([v, w]).

v v ’

Figure 2.4: The conductance for the edge [v, v0] is the ratio of the bold lengths

See Figure 2.4 Dubejko showed a disk triangulation graph is CP-hyperbolic orCP-parabolic as the tailored random walk is transient or recurrent

2.8 The type problem

We now come to our main motivating problem: How may we determine the VEL orEEL type of a given graph? We begin with the constant degree packings of Figure 2.2

If the graph has constant degree five, it is combinatorially forced to close up to asphere The constant degree six packing is the penny packing and corresponds tothe tiling of the plane by euclidean triangles The elements of euclidean geometry lie

at the core of this tiling – triangles have three sides, every triangle has angle sum

π, and π

3 = 2π

6 , so it is six euclidean equilateral triangles that must meet to get anangle of 2π going around a vertex If we try to get seven equilateral triangles to meetcoherently, we need smaller angles Hyperbolic geometry permits equilateral triangleswith any angle sum less than π, and so the constant degree n packings for n > 6 are

of hyperbolic type

It is useful to think in terms of angles, but it is difficult to see where the anglesare in a graph We instead appeal to another difference between hyperbolic and

Figure 2.5: Packing with alternating generations of degree 6 and 7 Hyperbolic orparabolic?

euclidean geometries – the growth of disks Whereas euclidean disks accrue area withthe square of the radius, the area of hyperbolic disks grow exponentially with radius.The discrete analog is to look at how may vertices lie within a fixed combinatorialdistance from the base vertex Indeed, recall from our definition of vertex extremallength that area was defined by an assignment of measures to the vertices

Consider again the constant degree six and seven graphs Choose a base vertex

v and denote the n-spheres about v of these graphs by S6(n) and S7(n), the set ofvertices for which the shortest vertex path to v0 has length n It is easy to see that

S6(n + 1) = S6(n) + 6 and so the combinatorial disk D6(n) = ∪j≤nS6(j) contains

nXj=0

|S6(n)| =

nXj=06n ≈ 6n2

vertices In the degree seven case, however, the number of vertices added at eachgeneration is not constant and depends on the size of the previous generation, makingthe growth of disks exponential (If you still need convincing that the constant degreeseven graph doesn’t belong in the euclidean plane, try drawing it.)

This gives us some properties to consider when addressing the type problem for acomplicated graph Our first new construction is to alternate generations of degreessix and seven as in Figure 2.5 The degree six generations want the graph to be

parabolic and the degree seven generations want it to be hyperbolic Who wins?Ryan Siders [Sid98] used electric network methods to solve the type problem forgraphs like this Not only is the graph of Figure 2.5 hyperbolic, but to make itparabolic one would have to include superexponentially more degree six generationsthan degree seven.

Vertex growth clearly is significant, but type problems get harder when we losethe symmetry of the graphs discussed so far Asymmetries in the graph can overcomethe effects of vertex growth, but there are some results connecting growth to type Heand Schramm [HS95] show that if all but finitely many vertices of a disk triangulationgraph have degree six, then the graph is parabolic On the other hand, the graph ishyperbolic if

!

> 6,with W and W0 finite, connected, and non-empty sets of vertices They alsoconjectured the following, since proved by Andrew Repp [Rep01] Let G be a disktriangulation graph with bounded degree As before, define the disk D(n) to be theset of vertices a combinatorial distance at most n from some fixed vertex v0 Definethe sequence

an= Xv∈D(n)(deg(v) − 6)

If this sequence is bounded, then G is parabolic For more results regarding vertexgrowth in bounded degree graphs via random walk methods, see [McC98] and[McC96]

The purpose of the next two chapters is to develop theorems that address thetype problem beyond the situations covered in this section We also present someexamples illustrating the limits of these theorems By the end of the discussion, weshould have not only an ample tool set for solving type problems, but also a bettersense of what makes them so tricky We then use these and other tools to explore theconnection between circle packings and equilateral surfaces in Chapter 5

CHAPTER 3 BOUNDED REFINEMENTS

The goal of this chapter is to develop combinatorial extremal length for graphsand prove our featured theorem, which says that if edges and vertices are added to agraph by a sufficiently reasonable process, then the change in the extremal length isbounded in a way that does not depend on the graph An immediate consequence isthat these processes preserve combinatorial type

3.1 Combinatorial extremal length

We begin with the general definitions for combinatorial extremal length We applythese definitions to graphs in Section 3.4 where we recover the definitions of vertexand edge extremal length from the previous chapter The reader is encouraged tokeep classical extremal length in mind for comparison

Let X be a non-empty set and Γ a non-empty collection of subsets of X A metric

on X is a function m : X → [0, ∞) The value m(x) may be called the m-weight orm-measure of x The area of m is

area(m) = X

x∈Xm(x)2,

and a metric is called admissible if it has finite area Let M(X) = {m : area(m) < ∞}

be the set of admissible metrics on X For A ⊂ X, define its m-length to be Lm(A) =P

x∈Am(x), and for a collection Γ of subsets of X, define Lm(Γ) = infA∈ΓLm(A) andthe extremal length

EL(Γ) = sup

m∈M(X)

 Lm(Γ)2area(m)



We say Γ is hyperbolic if EL(Γ) is finite and parabolic if EL(Γ) is infinite When theset Γ is clear from the context (and it usually will be), we refer to EL(X) = EL(Γ)

Note that scaling the metric does not change the quantity maximized by extremallength This fact allows us to assume that our metrics are always normalized to havearea one.

Proposition 3.1 (scale invariance) If µ is an admissible metric on X, Γ is acollection of subsets of X, and ν is the metric ν(x) = kµ(x) (k > 0), then

infγ∈ΓP

v∈γν(v)2P

x∈Xν(x)2 =

infγ∈ΓP

v∈γkµ(v)2P

Lµ(Γ)2area(µ)

Our next property is used extensively in the sequel Compare with its classicalcounterpart Theorem 2.1

Lemma 3.2 (monotonicity) Suppose Γ and Γ0 are collections of subsets of X withthe property that for every γ ∈ Γ there is a γ0 ∈ Γ0 such that γ0 ⊂ γ (In particular,this holds if Γ ⊂ Γ0.) Then EL(Γ0) ≤ EL(Γ)

Proof For each γ ∈ Γ, let γ0 ∈ Γ0 such that γ0 ⊂ γ For any metric m ∈ M(X), wehave



≥ supm∈M(X)

 Lm(Γ0)2area(m)



= EL(Γ0)

3.2 Extremal metrics

An extremal metric for Γ is an admissible metric µ for which EL(Γ) = Lµ (Γ) 2

area(µ) Inthe case EL(Γ) = ∞, an extremal metric has finite area and all elements of Γ haveinfinite length The following lemmas show such a metric always exists The existence

of a metric realizing the extremal length is of vital importance

Extremal metrics have a nifty geometric interpretation when X is finite Wesample the technique with the following lemma and proof from [Can94] See also[HS95]

Lemma 3.3 If X is finite and Γ is a collection of subsets of X, then EL(Γ) < ∞and there is an admissible metric m with area(m) = 1 and Lm(Γ)2 = EL(Γ)

Proof

Since X = {x1, , xn} is finite, we may view a unit area metric as an element

of the unit sphere in Rn That is, we may take a metric µ to be a vector

~µ = hµ(x1), , µ(xn)i, so that area(µ) = k~µk2 = 1 With this vector interpretation,

we may associate a path γ ⊂ X to the vector ~γ = hp1, , pni, where pi represents thenumber of appearances xi makes in γ Then Lµ(γ) = ~γ · ~µ is a continuous function of

µ for every γ Since the sphere is compact, its image under this function is boundedfor each γ and hence globally bounded over Γ Thus EL(Γ) < ∞

Since Lµ(γ) is continuous, infγ∈ΓLµ(γ)2 is also continuous Compactness againguarantees the supremum of this function is attained for some unit area metric m.This is the extremal metric

Cannon also shows in [Can94] that the extremal metric is unique up to scaling

We can also construct an extremal metric for infinite X and EL(Γ) = ∞

Lemma 3.4 If EL(Γ) = ∞, then there is a metric µ such that Lµ(Γ) = ∞ andarea(µ) = 1

Proof Suppose

EL(Γ) = sup

m∈M(X)

 Lm(Γ)2area(m)



= ∞

Then there is a sequence of unit area metrics m1, m2, · · · ∈ M(X) such that

Lm k(Γ) > k for all k Define

µ(x) =

∞Xk=1

mk(x)2

k2

! ∞Xk=1

1

k2

!

= π26Xx∈X

∞Xk=1

mk(x)2

k2 = π

26

∞Xk=1Xx∈X

1

k2Xx∈X

mk(x)2 = π

26

∞Xk=1

1

k2 area(mk) = π

26

∞Xk=1

mk(x)

k2 =

∞Xk=1Xx∈γ

mk(x)

k2

=

∞Xk=1

1

k2Xx∈γ

mk(x) ≥

∞Xk=1

1

k2Lmk(Γ) >

∞Xk=1

Let f and g be positive real-valued functions with domain Υ and k ≥ 1 Wesay the functions are k-comparable if 1

kg(υ) ≤ f (υ) ≤ kg(υ) for all υ ∈ Υ f and

g are comparable if they are k-comparable for some k ≥ 1 It is easy to verify thatcomparability defines an equivalence relation, and this is the relation we seek forextremal length, as exemplified by the following lemma

Lemma 3.5 Let X and Y be sets with ΓX ⊂ P(X), ΓY ⊂ P(Y ) Let {Ai}∞

i=0 and{Bi}∞

i=0 be collections of finite subsets of X and Y , respectively, and Γi

Y) are comparable (taken as functions of i)

Then EL(ΓX) = ∞ if and only if EL(ΓY) = ∞

Proof Suppose EL(ΓX) = ∞ with parabolic extremal metric µ as guaranteed byLemma 3.4 Define µi to be the restriction of µ to Ai and note that area(µi) ≤area(µ) = 1 Choose any N > 0 We show EL(ΓY) > N , implying EL(ΓY) = ∞.Since X is parabolic, every element γ ∈ ΓX has infinite µ-length Choose k > 0

so that k EL(ΓiX) ≤ EL(ΓiY) for all i > 0 All paths in ΓX have infinite µ-length, and

so for any given path γ ∈ ΓX there is an jγ > 0 so that Lµj(γ ∩ Aj) > qN

Lm(ΓjX)2area(m) = k EL(Γ

j

X) ≤ EL(ΓjY) ≤ EL(Y )

The last inequality is a direct consequence of monotonicity (Theorem 3.2) The proof

is completed by repeating the argument with the roles of X and Y reversed

3.3 Shadow paths

Most of the theorems of this chapter are about controlling the combinatorialextremal length of a set X that is related to some other set X0 whose extremallength is known We codify our technique in the following lemma

Lemma 3.6 Let X0 be a set, Γ0 a collection of subsets of X0, and µ0 an extremalmetric for (X0, Γ0) Suppose there is a set X, a collection Γ of subsets of X, a metric

µ on X, and constants C, D > 0 with the following properties:

EL(Γ) = sup

m∈M(X 0 )

infγ∈ΓLm(γ)2area(m) ≥

infγ∈ΓLµ(γ)2area(µ)

≥ infγ0∈Γ#(D · Lµ0(γ0))

2

2infγ 0 ∈Γ 0Lµ 0(γ0)2area(µ)

to construct a new metric µ on X satisfying the assumptions of Lemma 3.6, meaning

we always have two things to control: area and path length The trick is to constructthe metric so that the constants C and D depend on as little as possible

For x ∈ X, µ(x) is assigned a value of µ(x0) for some x0 ∈ X0 In this sense, eachelement x ∈ X has a corresponding element x0 ∈ X0 that prescribes its measure Theconstant C is a bound on the number of elements in X to which an element of X0may be assigned

For a path γ in Γ, we must guarantee a path in γ0 ∈ Γ0 whose µ0-length is lessthan D1 times the µ-length of γ The paths γ and γ0 naturally correspond As γ bobsand weaves through X, γ0 will “shadow” its movement in X0 and have similar length

These two conditions are at odds We need to choose µ carefully so that pathsare sufficiently long, but so that the area stays sufficiently small.

Our ultimate goal is comparability of the extremal lengths of two sets A and B.This requires applying Lemma 3.6 twice, with A and B alternating the roles of X and

X0 The metric constructions for the two proofs generally have little to do with oneanother The extremal lengths of A and B are shown to be k-comparable for some kdepending only on the constants in the lemma

3.4 Extremal length of graphs

Our definition of combinatorial extremal length is very general We now explorehow to connect combinatorial extremal length to graphs The reader is referred toSection 3.4 for the requisite graph theoretic definitions

Recall that we may apply the definition of combinatorial extremal length to agraph G = (V, E) in two ways, according to whether the metric assigns values tothe set of vertices or to the set of edges We say a planar graph is a combinatorialannulus or an annular complex if the graph is the 1-skeleton of topological annulus inthe plane, i.e a region A ⊂ C whose boundary consists of two disjoint Jordan curves

C1, C2 such that the bounded component of C \ C2 contains the bounded component

of C \ C1

Let G be a combinatorial annulus and let ΓV = ΓV(G) be the set of vertex pathsconnecting the two boundary components of G Define ΓE = ΓE(G) similarly foredge paths Define the vertex extremal length VEL(G) = EL(ΓV), and the edgeextremal length EEL(G) = EL(ΓE) Vertex and edge extremal length play the samerole for combinatorial annuli as classical extremal length does for conformal annuli.For example, we obtain the following estimate whose classical analog is known asRengel’s inequality

Theorem 3.7 (discrete Rengel’s inequality) Let G = (V, E) be a finite combinatorialannulus Then

VEL(G) ≥ RV(G)

2

|V |

2 The RW type, electrical type, and EEL type of G are the same

3 If G has bounded degree, then all described forms of combinatorial type. .. equilateral type of G to be the classical conformal type

of |G|eq

Equilateral type is defined as the classical type of a surface that is describedpurely combinatorially... bounded degree, then all described forms of combinatorial type –electrical type, RW type, CP type, VEL type, EEL type, and EQ type – arethe same

4 There is a graph that is EEL-hyperbolic and