Characterizing the polyhedral graphs with positive combinatorial curvature

Paul Richard Oldridge

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the Department of Computer Science

2017

c

⃝ Paul Richard Oldridge 2017 University of Victoria

Characterizing the Polyhedral Graphs with

Positive Combinatorial Curvature

Paul Richard Oldridge

Supervisory Committee

Dr. Wendy Myrvold, Supervisor Department of Computer Science

Dr. Frank Ruskey, Departmental Member Department of Computer Science

The combinatorial curvature of a vertex v in 2-cell embedded graph G is defined as Φ(v) = 1− deg(v) 2 + ∑ f∈F (v) 1 |f|

where F (v) is the set of faces that are incident to v. G is called PCC if every vertex of G has strictly positive combinatorial curvature and the graph is not a prism or antiprism. In this thesis it is shown that the maximum order of a 3-regular PCC graph is 132 and the 3-regular PCC graphs which match that bound are enumerated. A new PCC graph with two 39-faces and 208 vertices is constructed, matching the number of vertices of the largest PCC graphs discovered by Nicholson and Sneddon. A conjecture that there are no PCC graphs with faces of size larger than 39 is made, along with a proof that if there are no faces of size larger than 122, then there is an upper bound of 244 on the order of PCC graphs.

Table of Contents

Supervisory Committee ii

Abstract iii

Table of Contents iv

List of Tables vi

List of Figures vii

1 Introduction 1

1.1 Background . . . 4 1.2 Outline . . . 5

2 PCC Vertex Types 6

3 An ILP for PCC graphs 8

4 An Upper Bound on 3-regular PCC Face Size 15

5 3-regular PCC Graphs of Maximum Order 22

5.1 Optimal 3-Regular PCC ILP Solutions . . . 24 5.2 Maximum Order 3-Regular PCC Graphs . . . 26 5.3 Projective Plane . . . 29

6 PCC Graphs with Large Faces 30

6.1 Medium Faces . . . 30 6.2 Big Faces . . . 35 6.3 Monster Faces . . . 38

9 Bounding the General Case 43

10 Conclusions 48

List of Tables

2.1 All vertex types with positive combinatorial curvature . . . 7

3.1 PCC∗_{3,4,...,k} upper bound obtained from ILP . . . 10

3.2 Number of k-cycle sequence types for k from 3 to 41 . . . . 12

3.3 PCC∗_{3,4,...,k} upper bound obtained from ILP with angles . . . 14

5.1 132-vertex solutions to the 3-regular ILP . . . 24

7.1 PCC∗_{4,7,9} ILP solutions . . . 40

7.2 PCC∗_{3,10,14} ILP solutions . . . 41

9.1 General ILP solution . . . 44

List of Figures

1.1 A graph G and an induced subgraph G− v3. . . 2

1.2 A prism and an antiprism. . . 3

1.3 A projective prism and a projective antiprism. . . 4

3.1 A 4-cycle and vertex with sequence type (3, 4, 3, 11) that share an angle of type A4(3, 3). . . . 13

4.1 Case 1: Neither u nor v is incident to a triangle. . . . 17

4.2 Case 2: Only u is incident to a triangle. . . . 17

4.3 Case 3: u and v are both incident to same triangle. . . . 18

4.4 Case 4: u and v are each incident to different triangles. . . . 20

5.1 Switching the colour of a white face to red. . . 23

5.2 The cycle type subsequence 10, 3, 10 on a 10-face . . . . 25

5.3 Great Rhombicosidodecahedron with embedded Icosahedron . . . . 27

5.4 Five 3-regular PCC graphs with 132 vertices . . . 28

5.5 Maximum order 3-regular PCC graphs in the projective plane . . . 29

6.1 G20 and G22. . . 31

6.2 Two motifs X and Y . . . . 31

6.3 A PCC graph with two faces of size 23. . . 32

6.4 The number of PCC graphs constructed with two k-faces . . . 33

6.5 The 20 22-face PCC graphs that this construction yields . . . 34

6.6 A motif for constructing PC graphs with large faces. . . 35

6.7 A planar PCC graph with 208 vertices and two 39-faces. . . 36

6.8 A projective planar PCC graph with 104 vertices and a 39-face. . . 37

7.1 A maximum order PCC∗_{4,7,9} graph . . . 40

9.1 4-face motifs . . . 43 9.2 4-faces with augmented cycle types . . . 44

CHAPTER 1

Introduction

An undirected graph is an ordered pair G = (V, E), where V is a set of

ver-tices, and E is a set of edges where each edge corresponds to an unordered

pair of vertices from V . We will use the notation V (G) to denote the vertex set, and E(G) to denote the edge set of a graph G. A subgraph H of G is a graph where V (H) ⊆ V (G) and E(H) ⊆ E(G). A vertex and edge are

incident to each other if the vertex is an endpoint of the edge. The

num-ber of edges that are incident to a vertex u is called the degree of u and is denoted deg(u). A graph G is called 3-regular if every vertex in V (G) has degree three. The subgraph of G induced by S ⊆ V (G), is denoted G[S], and is obtained by removing all the vertices from G which are not in S, along with their incident edges. The graph G− S is the subgraph of G induced by V (G)− S. Figure 1.1 shows an example of a graph G and the induced subgraph G− {v3}.

A path of length k, from v0 to vk, is an alternating sequence of vertices and

edges of the form v0, (v0, v1), v1, (v1, v2), v2, ..., vk−1, (vk−1, vk), vk. A graph is

connected if there is a path from u to v for all u, v ∈ V . A graph H is k-connected if for all sets S ⊂ V (H), with |S| ≤ k − 1, the graph H − S is

connected.

A planar graph is a graph that can be embedded in the plane, such that each vertex is a distinct point and each edge is a simple curve with endpoints on its two incident vertices. A polyhedral graph is a graph that is both planar

Figure 1.1: A graph G and an induced subgraph G− v3.

and 3-connected.

A projective planar graph is a graph that can be embedded in the projective plane, such that each vertex is a distinct point and each edge is a simple curve with endpoints on its two incident vertices. A projective polyhedral graph is a graph that is both projective planar and 3-connected.

A face is a maximal empty region in an embedding. A face F and vertex v are incident to each other if the vertex v is on the boundary of the face f . The size of a face F , denoted f =|F |, equals the number of vertices incident to F . The type of a vertex T (v), is the multiset containing the sizes of the faces that are incident to v.

The combinatorial curvature, herein called curvature, of a vertex v in an embedding is defined as:

Φ(v) = 1−deg(v) 2 + ∑ f∈T (v) 1 f

Prism Antiprism Figure 1.2: A prism and an antiprism.

If each face of an embedding is regarded as a regular polygon then the sum of the angles given by each face incident to a vertex v is given by 2π− 2πΦ(v). A polyhedral or projective polyhedral graph is said to be positively curved if it has strictly positive curvature at each of its vertices. There exist two in-finite classes of positively curved polyhedral graphs: prisms and antiprisms. An example of a prism and an antiprism is given in Figure 1.1. A prism is the skeleton of a geometric prism. An antiprism is a prism with k ad-ditional edges, dividing each square face into two triangles, such that each vertex has degree four. Sun and Yu [1] proved that the set of 3-regular graphs with everywhere strictly positive curvature that are not prisms nor projective prisms is finite.

We denote the set of polyhedral graphs that have planar or projective planar embeddings with everywhere strictly positive curvature, excluding prisms, antiprisms, and their projective analogues, as PCC. For a set of integers S, we will use the notation PCC∗S to denote the subset of PCC consisting of

graphs which only have faces with sizes in S. We will use the notation PCCS

to denote the subset of PCC∗S consisting of graphs which have at least one

Projective Prism Projective Antiprism Figure 1.3: A projective prism and a projective antiprism.

1.1

Background

In 2001, Higuchi [2] conjectured that a graph which has every vertex with strictly positive curvature cannot be an infinite graph. Sun and Yu con-firmed Higuchi’s conjecture for all 3-regular polyhedral graphs [1], and then in 2004, DeVos and Mohar confirmed that the conjecture is true for all poly-hedral graphs [3].

DeVos and Mohar [3] showed that the set PCC (which excludes the infinite classes of prisms and antiprisms) is finite. They proposed the question of determining the maximum order of a PCC graph and gave a lower bound of of 120, which is realized by a great rhombicosidodecahedron, and an upper bound of 3444. Chen and Chen [4] independently obtained the same upper bound.

In 2005, Reti, Bitay, and Kosztol´anyi [3] improved the lower bound to 138 [5] and conjectured that there are no PCC graphs having faces larger than 19

20 and 22. In Chapter 6 it is shown that there exist other PCC graphs with faces of size greater than 19. They also found a 3-regular graph with 132 vertices and conjectured that it is a largest 3-regular graph in PCC_{4,6,11}. This conjecture is proven in Chapter 5.

Recently Zhang claimed to decrease the upper bound to 579 [6], by showing that any positively curved graph with at least 580 vertices must be a prism or antiprism. In Chapter 8, it is shown that Zhang’s proof is not correct. Nicholson and Sneddon improved the lower bound by finding some large non-3-regular PCC graphs [7], the largest of which has 208 vertices.

Fowler, Nikoli´c, De Los Reyes, and Myrvold [8] enumerated the PCC fullerenes.

1.2

Outline

In Chapter 2, the vertex types for PCC graphs are characterized. Chapter 3 formulates an integer linear programming problem (ILP) that gives an upper bound on the order of PCC graphs. An upper bound on the size of faces in a 3-regular PCC graph is given in Chapter 4. In Chapter 5 an upper bound on the order of 3-regular PCC graphs is computed by using the ILP formulation from Chapter 3 with the face size bound from Chapter 4. Chapter 6 discusses PCC graphs with large faces, including a PCC graph with two 39-faces and a 3-regular PCC graph with two 22-faces. The graph with two 39-faces has the same number of vertices (208) as the current lower bound due to Nicholson and Sneddon [7], but uses none of the same vertex types. In Chapter 8 previous work on the upper bound for the order of PCC graphs is discussed. Chapter 9 consists of new work towards a general upper bound for the maximum number of vertices in a PCC graph.

CHAPTER 2

PCC Vertex Types

Recall that the formula for the combinatorial curvature of a vertex v is:

Φ(v) = 1− deg(v) 2 + ∑ f∈T (v) 1 f . (2.1)

The curvature of a degree d vertex v is maximized when all the faces incident to v have three vertices. A vertex of degree d surrounded by d 3-faces has curvature 1− d₂ + d₃ = 1− d₆. Since this is less than or equal to zero for

d ≥ 6, there are no vertex types of degree six or higher that yield positive

combinatorial curvature. Table 2.1 shows the vertex types with positive curvature. Theorem 2.1 can be derived from Equation 2.1 and the Euler characteristics for the projective plane and sphere.

Theorem 2.1 Let G be a PCC graph. The sum of the curvatures of the vertices of G equals 2X where X equals 1 if G is embedded in the projective plane and 2 if G is embedded in the plane. □

Vertex Type Curvature {3, 3, k} 3≤ k 1₆ + 1_k {3, 4, k} 4≤ k ₁₂1 +1_k {3, 5, k} 5≤ k ₃₀1 +1_k {3, 6, k} 6≤ k 1_k {3, 7, k} 7≤ k ≤ 41 1_k − ₄₂1 {3, 8, k} 8≤ k ≤ 23 1_k − ₂₄1 {3, 9, k} 9≤ k ≤ 17 1_k − ₁₈1 {3, 10, k} 10 ≤ k ≤ 14 1 k − 1 15 {3, 11, k} 11 ≤ k ≤ 13 1 k − 5 66 {4, 4, k} 4≤ k 1_k {4, 5, k} 5≤ k ≤ 19 1_k − ₂₀1 {4, 6, k} 6≤ k ≤ 11 1_k − ₁₂1 {4, 7, k} 7≤ k ≤ 9 1_k − ₂₈3 {5, 5, k} 5≤ k ≤ 9 1_k − ₁₀1 {5, 6, k} 6≤ k ≤ 7 1_k − ₁₅2 Degree 3

Vertex Type Curvature

{3, 3, 3, k} 3≤ k 1_k

{3, 3, 4, k} 4≤ k ≤ 11 1_k− ₁₂1

{3, 3, 5, k} 5≤ k ≤ 7 1_k− ₁₅2

{3, 4, 4, k} 4≤ k ≤ 5 1_k− 1₆ Degree 4

Vertex Type Curvature

{3, 3, 3, 3, k} 3 ≤ k ≤ 5 1

k−

1 6 Degree 5

CHAPTER 3

An ILP for PCC graphs

Given a set S of face sizes, an integer linear programming problem (ILP) that gives an upper bound on the maximum number of vertices of a graph in PCC∗S can be formulated as described in this section. The viable vertex

types for S are the vertex types that consist of face sizes in S and also have strictly positive combinatorial curvature. The complete set of vertex types that have positive curvature for face sizes at most 41 is included in the ap-pendix.

A vertex sequence type is defined for each cyclic sequence of face sizes that correspond to a vertex type with strictly positive curvature. Two vertex se-quence types are equivalent if one can be obtained from the other by either rotating the sequence, or flipping it then rotating. The canonical

represen-tative is chosen to be the lexicographic minimum sequence obtained using

these operations. By this definition, a vertex sequence type for a degree three vertex of type {a, b, c} is the same as any of the other vertex sequence types with the same multiset of values. The canonical representative has

a≤ b ≤ c. Two vertices of degree four can have the same multisets of values

but correspond to different vertex sequence types. For example, (3, 3, 5, 7) is different from (3, 5, 3, 7). The only valid vertex sequence types of degree five have four 3’s and a value x where x is between three and five and such a vertex sequence type has canonical representative (3, 3, 3, 3, x). The set of canonical representatives for vertex sequence types having all face sizes in S is denoted TS.

The ILP is designed so that it determines an upper bound on the value for

n for any PCC graph having all its face sizes in S. The objective is to

max-imize n. The variables used are:

1. The number of vertices: n 2. The number of edges: m 3. The number of faces: f

4. The number of vertices of each vertex sequence type t∈ TS: xt

5. For each face size s∈ S, the number of faces of size s: fs

The constraints are as follows.

Constraint 3.1 asserts that every vertex has a vertex sequence type:

n = ∑

t∈TS

xt . (3.1)

Constraint 3.2 is used to count the number of edges m in a graph corre-sponding to the solution to the ILP. The degree of a vertex with vertex sequence type t is |t|. The sum of the degrees is two times the number of edges.

2m = ∑

t∈TS

|t|xt . (3.2)

Constraint 3.3 is used to count the number of faces f in a graph correspond-ing to a solution to the ILP. The number of faces is the sum of the number of faces of each size:

f = ∑

s∈S

fs . (3.3)

k Upper Bound Vertex Sequence Type 3 12 (3, 3, 3, 3, 3) 4 24 (3, 3, 3, 3, 4) 6 60 (3, 3, 3, 3, 5) 8 210 (5, 6, 7) 12 504 (4, 7, 9) 40 1716 (3, 11, 13) 1722 3444 (3, 7, 41)

Table 3.1: PCC∗_{3,4,...,k} upper bound obtained from ILP

one for embeddings in the projective plane and two for the sphere. Except where specified solutions to the ILP discussed throughout this thesis are for

X = 2. Euler’s formula is added as Constraint 3.4:

X = n + f − m . (3.4)

Let m(s, t) denote the multiplicity of s in the vertex sequence type t. For each face size s∈ S:

sfs =

∑

t∈TS

m(s, t) xt . (3.5)

The ILP problem described thus far was generated by a computer program for various face size sets and solved using SCIP [9]. The results for the face size set S = {3, 4, . . . , k} for various values for k are given in Table 3.1 and are an upper bound on PCC∗S. Note that for each k the bound corresponds

to the bound from the appendix for the first vertex type that has all face sizes at most k.

odd number, such a polyhedron cannot be constructed because it implies that the face sizes around the 41-faces are alternately 3 and 7-faces. This does not work for a 41-face because 41 is an odd number. Hence, some vertex on the 41-face is not a (3, 7, 41)-vertex. One alternative is to include one (3, 3, 41)-vertex. In order to guard against numerical solutions like this that are not realizable, some extra equations described below are added to the ILP.

For each face F of size k, consider the cyclic sequence of the k face sizes of the faces sharing an edge with F . For a k-cycle sequence of the form

f1, f2, . . . , fk, the vertex-type set of the cyclic sequence consists of a

multi-set containing the canonical representatives of the vertex types (k, fi, fi+1)

for i = 1, 2, . . . k where the addition is done modulo k. The canonical

repre-sentative of a k-cycle sequence is defined to be the lexicographic minimum

of all the k-cycle sequences having the same multi-set of vertex types.

A k-cycle sequence is not canonical if a lexicographically smaller one can be obtained by rotating, or flipping and then rotating the sequence values. But there are also cases such as the 6-cycle sequence 3, 3, 4, 3, 3, 4 whose vertex type multi-set contains two (3, 3, 6) types and four (3, 4, 6) types that are not canonical because for example 3, 3, 3, 4, 3, 4 is lexicographically smaller and it has the same vertex-type multi-set.

For each canonical k-cycle sequence type t, a variable yt is included in the

ILP representing the number of faces in the solution that are in the same equivalence class as t. The set of canonical k-cycle sequence types is denoted by Ck. The number of canonical k-cycle sequence types for k ranging from

k # sequence types k # k # 3 771 16 19248 29 17817 4 3014 17 27036 30 21013 5 2916 18 20096 31 24073 6 14240 19 26894 32 28102 7 22295 20 9010 33 31978 8 17555 21 11012 34 36994 9 14990 22 14379 35 41839 10 11599 23 17382 36 48014 11 15577 24 7827 37 53999 12 8664 25 9192 38 61524 13 13163 26 11117 39 68839 14 14517 27 12937 40 77925 15 12680 28 15437 41 86780

Table 3.2: Number of k-cycle sequence types for k from 3 to 41

counts were verified using two different programs written independently by two programmers (Oldridge and Myrvold). The counts are not monotoni-cally increasing because for example, a type such as (3, 8, k) is only valid for

k ≤ 23.

The first set of new equations added to the ILP ensure that each face of size

k corresponds to one of the k-cycle sequence types. For each k ∈ S:

fk =

∑

t∈Ck

yt. (3.6)

A k-cycle sequence type of the form f1, f2, ..., fk, has k angles where angle i

addi-Figure 3.1: A 4-cycle and vertex with sequence type (3, 4, 3, 11) that share an angle of type A4(3, 3).

tion is done modulo k. The two types Ak(a, b) and Ak(b, a) are considered

to be equivalent to each other and the canonical representative has a≤ b.

The variable ak(a, b) is equal to the number of times there is an angle of type

Ak(a, b) on some k-cycle. The number of times each angle type appears when

enumerating angles over the faces of the graph is:

ak(a, b) =

∑

t∈Ck

ytm(Ak(a, b), t) (3.7)

where m(Ak(a, b), t) is the number of times that the k-cycle sequence type t

has an angle of type Ak(a, b).

The vertex types for the k-cycle sequence types are all degree three. This arises because the k-cycle sequence types are only taking into consideration the sizes of the faces that share an edge with a given face. The angle types can also be enumerated by considering the types of the vertices (where the vertex degrees can be three, four or five).

A degree d vertex that has vertex-type sequence f1, f2, ..., fd, has d angles

where the ith _{one is of type A}

fi(fi−1, fi+1) where the addition is modulo d.

k Upper Bound Vertex Sequence Types 3 12 (3, 3, 3, 3, 3) 4 24 (3, 3, 3, 3, 4) 6 60 (3, 4, 4, 5), (3, 4, 5, 4) or (5, 6, 6) 10 210 (3, 3, 5, 7), (3, 5, 3, 7), (5, 6, 7) 132 264 (3, 3, 4, 11), (3, 4, 3, 11), (4, 6, 11)

Table 3.3: PCC∗_{3,4,...,k} upper bound obtained from ILP with angles

the ones from the k-cycle sequence types:

ak(a, b) =

∑

t∈TS

xtm(Ak(a, b), t) (3.8)

where m(Ak(a, b), t) is the number of times that the vertex sequence type t

has an angle of type Ak(a, b).

For 3-regular graphs, it suffices to generate the k-cycle sequence constraints up to k equal to 23 because in Chapter 4, we prove that 3-regular PCC graphs have no faces of size 24 or more. For the general problem, the constraints were generated for face sizes up to 41.

CHAPTER 4

An Upper Bound on 3-regular PCC Face Size

In this chapter we will show that 3-regular PCC graphs have maximum face size at most 23.

Theorem 4.1 If G is a 3-regular PCC graph, then the faces of G have at most 23 vertices.

A proof by contradiction follows. Assume that some 3-regular PCC graph

G exists that has a maximum sized face B with b =|B| ≥ 24. Starting with

each vertex z of G having charge equal to its curvature Φ(z), the following discharging rules are used to send charge from the vertices of G to the edges of B:

Rule 1: If z is on B and B contains the path u, z, v, then z should discharge 1

2Φ(z) to each of (u, z) and (z, v).

Rule 2: If z is not on B, but is adjacent to exactly one vertex v on B and B contains the path u, v, w, then z should discharge 1₂Φ(z) to each of (u, v) and (v, w).

Rule 3: If z is not on B and is adjacent to two vertices v and w on B where the face B includes the path u, v, w, x, then z should discharge 1₂Φ(z) to (v, w) and 1₄Φ(z) to each of (u, v) and (w, x).

Rule 4: If z is not on B nor adjacent to any vertices on B, but is adjacent to a vertex y that is adjacent to two vertices v and w on B where the face

B includes the path u, v, w, x and (z, y) lies on two 7-faces, then z should

discharge 1₃Φ(z) to each of (u, v), (v, w), and (w, x).

Every vertex z ∈ V (G) discharges according to at most one rule since G is 3-connected. The proof considers the sum of the curvatures of the vertices in G and shows that is at least two, and is only equal to two if the graph is a prism. Since the curvature of a PCC graph sums to at most two, 3-regular PCC graphs have faces of size at most 23.

For each edge e of B, bΦ(e) is defined as the charge received from the dis-charging rule. The proof proceeds by verifying that every edge (u, v) of B receives at least 2_b charge and only has charge equal to 2_b if u and v are both vertices of type (4, 4, b). For an edge (u, v) of B, the vertex types of u and

v must be of the form {3, x, b} where 3 ≤ x ≤ 7, or {4, 4, b}.

Case 1: Neither u nor v is incident to a triangle [Figure 4.1].

Since u and v are not incident to a triangle they must both be of type

{4, 4, b}. The combinatorial curvature of a {4, 4, b} vertex is 1

b. By Rule 1,

vertices u and v each send _2b1 charge to e. Vertices w and x are also incident to two 4-faces, thus they are of the form{4, 4, g} and since B is a maximum sized face of G, g ≤ b. Therefore, by Rule 2, w and x each discharge at least

1

2b to e. In total bΦ(e) ≥ 4 1 2b =

2

b. Note that bΦ(e) is exactly

2

b only when

g = b.

Case 2: One of u and v is incident to a triangle and the other is not [Figure 4.2].

Figure 4.1: Case 1: Neither u nor v is incident to a triangle.

Figure 4.2: Case 2: Only u is incident to a triangle.

Without loss of generality, let u be the vertex that is incident to a triangle. Vertex v is not incident to a triangle, so it has type {4, 4, b} which has cur-vature 1

b. Vertex u is of type {3, 4, b} which has

1 12 +

1

b curvature. By Rule

1, vertices u and v each send half their charge to e, so together they send 1 2 ( (1 12+ 1 b) + 1 b ) = 1 24 + 1 b, which is at least 2

b since b ≥ 24. Note that the

vertices that are adjacent to u or v, but not on B, also send some charge to

e, so bΦ(e) must be strictly greater than 2_b.

Case 3: u and v are both incident to the same triangle [Figure 4.3].

Suppose that the two faces with edge (w, x) have sizes f1 =|F1| and f2 =|F2 and without loss of generality f1 ≤ f2. This case is split into two sub-cases. In each, it is shown that the charge received by the edge (u, v) is at least

1

b +

1 21.

Case 3.1: f1 ≤ 6. Vertices u and v send half their charge to the edge e = (u, v) by Rule 1 and vertex w sends half its charge by Rule 3. In total

Figure 4.3: Case 3: u and v are both incident to same triangle.

e receives 1₂(Φ(u) + Φ(v) + Φ(w)) charge. The vertex types that minimize the charge received by e are {3, 6, b} for u, {3, 7, b} for v, and {3, 6, 7} for

w; which have curvatures of 1_b, 1_b − ₄₂1, and 1₇ respectively. Thus e receives at least 1 2 ( 1 b + 1 b − 1 42+ 1 7 ) = 1 b + 5 84 > 1 b + 4 84 = 1 b + 1 21 charge.

Case 3.2: f1 = 7 (and hence f2 = 7). Vertices u and v send half their charge to the edge e = (u, v) by Rule 1, vertex w sends half its charge by Rule 3, and vertex x sends one third of its charge by Rule 4. In total e receives

1

2(Φ(u) + Φ(v) + Φ(w)) + 1

3Φ(x) charge. The vertex types that minimize the charge received by e are{3, 7, b} for u and v, {3, 7, 7} for w, and (4, 7, 7) for x; which have curvatures of 1_b − ₄₂1, 1₇ − ₄₂1 and ₂₈1 respectively. Thus e receives at least

1 2 ( Φ(u) + Φ(v) + Φ(w) ) + 1 3Φ(x) = 1 2 [ 2 ( 1 b − 1 42 ) + ( 1 7 − 1 42 )] + 1 3· 1 28 = 1 b − 1 42+ 1 2· 5 42 + 1 84 = 1 b + 1 21 charge.

In both sub-cases e receives at least 1_b + ₂₁1 which is greater than 2_b since

b ≥ 24.

Case 4: u and v are each incident to distinct triangles [Figure 4.4].

Without loss of generality consider only the charge that comes from the u side of the edge. That is the charge from vertices u, w, and x in Figure 4.4. The charge received by the edge e = (u, v) is at least double the minimum charge it must receive from u, w, and x.

Suppose that the two faces with edge (w, x) have sizes f1 =|F1| and f2 =|F2| and without loss of generality f1 ≤ f2. This case is split into two sub-cases. In each, it is shown that the charge received by the edge (u, v) is at least

1

b +

1

21 since it receives at least 1 2b +

1

42 from each side.

Case 4.1: F1 and F2 are not both 7-faces. Vertex u sends half its charge to the edge e = (u, v) by Rule 1 and vertex w sends one quarter of its charge by Rule 3. In total e receives at least 1₂Φ(u) + 1₄Φ(w) charge. The vertex types that minimize the charge received by e are {3, 7, b} for u and (3, 6, 7) for w, which have curvatures of 1_b −₄₂1, and 1₇ respectively. Thus e receives at least

Figure 4.4: Case 4: u and v are each incident to different triangles. 1 2 ( 1 b − 1 42 ) +1 4· 1 7 = 1 2b − 1 84+ 1 28 = 1 2b + 2 84 = 1 2b + 1 42

charge from the u side.

Case 4.2: F1 and F2 are both 7-faces. Vertex u sends half its charge to the edge e = (u, v) by Rule 1, vertex w sends one quarter of its charge by Rule 3, and x sends one third of its charge by Rule 4. In total e receives at least 1₂Φ(u) + 1₄Φ(w) + 1₃Φ(x) charge. The vertex types that minimize the charge received by e are {3, 7, b} for u, {3, 7, 7} for w, and {4, 7, 7} for x, which have curvatures of 1_b −₄₂1 , 1₇−₄₂1 , and ₂₈1 respectively. Thus e receives at least

1 2 ( 1 b − 1 42 ) + 1 4 ( 1 7 − 1 42 ) +1 3· 1 28 = 1 2b − 1 84+ 1 4· 5 42 + 1 84 = 1 2b + 1 4· 5 42 > 1 2b + 1 4· 4 42 = 1 2b + 1 42

charge from the u side.

In both sub-cases 4.1 and 4.2 e receives at least _2b1 +₄₂1 charge from each side and thus receives at least 1_b +₂₁1 total, which is greater than 2_b since b ≥ 24.

In every case the charge received by e is at least 2_b and it can only be equal to 2_b when u and v are both {4, 4, b} vertices (Case 1). Since there are b edges on the cycle B, the sum of all the attributed curvatures is at least two, which is a lower bound on the sum of the curvatures of V (G). By Theorem 2.1 the curvature of a polyhedral graph must sum to at most two, therefore every vertex of B must have the type {4, 4, b}. It follows that G is a prism and not a PCC graph which is a contradiction. Hence, there are no 3-regular PCC graphs with faces of size 24 or greater. □

CHAPTER 5

3-regular PCC Graphs of Maximum Order

Constraints can be added to the ILP problem defined in Chapter 3 to restrict solutions to interesting subclasses. For this chapter a constraint is added which asserts that all vertices have degree equal to three:

n = ∑

{t : t ∈ T and |t| = 3}

xt. (5.1)

With Constraint 5.1, the ILP gives an upper bound of 138 on the order of a 3-regular PCC graph. Theorem 5.1 is used to define some new constraints which will tighten the bound further.

Theorem 5.1 If every face of a 3-regular graph, embedded in either the plane or the projective plane so that no faces have repeated vertices, is coloured either red or white (adjacent faces can have the same colour), then the number of vertices incident to two or more red faces is even.

Proof. A 3-regular graph embedding, with all faces coloured white forms the basis for a proof by induction on the number of faces coloured red. The basis has an even number of vertices incident to two or more red faces (zero). Assume that all 3-regular graph embeddings with k red faces have an even number of vertices incident to two or more red faces. Consider the cyclic sequence of colours of the faces surrounding a white face F , in some 3-regular graph embedding with k red faces, using w to denote white and r to denote red. For example, the central face in Figure 5.1 has the sequence cyclic wwrrrwrwrr starting with F1 and proceeding clockwise. Since these

Figure 5.1: Switching the colour of a white face to red.

are cyclic sequences, there must be a transition back from white to red for every transition from red to white, thus there is an even number of tran-sitions from red to white or vice-versa. At, and only at, those trantran-sitions there is a vertex that is currently incident to one red face and will gain the property of being incident to two or more red faces if the colour of F is switched to red. Hence, switching the colour of a white face F to red in a graph embedding with k red faces results in a graph with k + 1 red faces and increases the number of vertices that are incident to two or more red faces by an even amount. Therefore by induction, all 3-regular graph embeddings with no faces having repeated vertices must have an even number of vertices incident to two or more red faces. □

Corollary 5.2 is obtained from Theorem 5.1 by considering an embedding of a 3-regular graph and colouring every face of size s red and all the other faces white.

Corollary 5.2 For every integer s, the number of vertices incident to two or more faces of size s, in a 3-regular planar or projective planar embedding with no faces having repeated vertices G, is even.

Let Dr = {t ∈ T : |t| = 3 and t contains at least 2 r’s}. The following

Solution f3 f4 f6 f10 f11 f14

1 0 36 20 0 12 0

2 44 0 0 18 0 6

3 22 18 10 9 6 3

Table 5.1: 132-vertex solutions to the 3-regular ILP

ensure that the integer solutions to the problem respect Corollary 5.2. For each integer r in 3≤ r ≤ 11:

∑

t∈Dr

xt= 2er . (5.2)

5.1

Optimal 3-Regular PCC ILP Solutions

After adding the constraints 5.1 and 5.2 to the ILP from Chapter 3, the 3-regular upper bound obtained from the ILP is 132. This matches the lower bound for the 3-regular subclass realized by the 3-regular PCC graph which was discovered by Reti, Bitay, and Kosztol´anyi [5]. They conjectured that this is a maximum order graph in PCC(4,6,11). That conjecture is true, as is the stronger statement in Theorem 5.3.

Theorem 5.3 The 3-regular PCC graph with 132 vertices discovered by Reti, Bitay, and Kosztol´anyi [5] is a maximum order 3-regular PCC graph.

SCIP gives multiple solutions to the ILP that yield n = 132. Table 5.1 categorizes three solutions by the face sizes used.

Solution 1 corresponds to the graph discovered by Reti, Bitay, and Kosz-tol´anyi. Solutions 2 and 3 both use 10-faces with cycle sequence types con-taining the subsequence 10, 3, 10, but no 3-faces with cycle sequence type

A cycle sequence type t is said to have k occurrences of a subsequence q, if every cycle sequence type that has t as its canonical representative accord-ing to the rules in Chapter 3 has the subsequence q startaccord-ing in at least k different positions (cyclically).

The cycle sequence type 3,10,3,10,3,10,3,10,3,10 has five occurrences of the subsequence 10, 3, 10 and the cycle sequence type 3,10,3,10,3,10,3,10,3,10 has three occurrences of the subsequence. The cycle sequence type 3,10,3, 10,3,10,3,14,3,14 has one occurrence of the subsequence, since it is the canon-ical representative of 3,10,3,10,3,14,3,10,3,14, which only has the subse-quence 10, 3, 10 in one place.

Figure 5.2: The cycle type subsequence 10, 3, 10 on a 10-face

Figure 5.2 shows part of a 10-face that has a 10, 3, 10 subsequence in its cycle sequence type. For every three of these occurrences of the subsequence 10, 3, 10, there must be at least one triangle that has the cycle size sequence 10, 10, 10, which leads to the following constraint:

5y3,10,3,10,3,10,3,10,3,10 +3y3,10,3,10,3,10,3,10,3,14

+y3,10,3,10,3,10,3,14,3,14 ≤ 3y10,10,10 .

Adding Constraint 5.3 along with a constraint requiring that there is at least one 10-face results in an upper bound of 130. Therefore, a 3-regular PCC graph with 132 vertices does not have any 10-faces and graphs realizing the 132 bound must correspond to Solution 1.

5.2

Maximum Order 3-Regular PCC Graphs

Reti, Bitay, and Kosztol´anyi [5] described how to construct a 3-regular PCC graph embedded in the plane with 132 vertices by choosing six 4-faces of the Great Rhombicosidodecahedron, incident to distinct 10-faces, and for each of those 4-faces, subdividing the two edges which lie on the boundary of a 10-face and adding an edge which has the newly added vertices as its end-points. Such a construction corresponds to choosing a perfect matching of an Icosahedron embedded in the Great Rhombicosidodecahedron such that the vertices are embedded in the 10-faces of the Great Rhombicosidodeca-hedron and there is an edge corresponding to each place that two 10-faces share a common 4-face. See Figure 5.3. Up to isomorphism there are five perfect matchings of the Icosahedron [10]. The five 3-regular PCC graphs with 132 vertices corresponding to these are shown in Figure 5.4

Any graph realizing Solution 1 must correspond to one of the five graphs in Figure 5.4. One way to prove this is to consider just the 11-faces. Because an odd size face cannot have an alternating sequence of 4 and 6-cycles around it. Each 11-face must have an odd number of vertices of type{4, 4, 11}, thus each 11-face has at least one such vertex, and since there are twelve 11-faces and also twelve vertices of type{4, 4, 11}, each 11-face must have exactly one vertex of type {4, 4, 11}. If a vertex u has type {4, 4, 11} then it is incident to an edge that lies on two 4-faces and the other endpoint of that edge must also be of a vertex that is incident to two 4-faces, which is a{4, 4, 11}.

Figure 5.3: Great Rhombicosidodecahedron with embedded Icosahedron

Therefore the {4, 4, 11} vertices are in pairs and there are six edges that lie on two 4-faces. Removing such edges and smoothing their endpoints results in a graph with 30 isolated 4-cycles, 20 isolated 6-cycles, and 12 isolated 10-cycles. That can only be realized by the Great Rhombicosidodecahedron.

5.3

Projective Plane

In the projective plane X = 1 and SCIP gives an upper bound of 66. This is realized by the graphs in Figure 5.5.

CHAPTER 6

PCC Graphs with Large Faces

It is interesting to study PCC graphs with large faces; particularly since large faces contribute very small amounts to the curvature for each of their vertices. R´eti, Bitay, and Kosztol´anyi [5] made the following two conjectures:

Conjecture 6.1 There are no PCC graphs having faces of size greater than 19, except for G20 and G22, pictured in Figure 6.1, which have faces of size 20 and 22 respectively.

Conjecture 6.2 The minimum curvature at a vertex in a PCC graph is at least 1

380, the curvature of a {4, 5, 19}-vertex.

Several counterexamples for Conjecture 6.2 were given by Nicholson and Sneddon, and several counterexamples for Conjecture 6.1 are given herein. We call a face medium if it has between 14 and 23 vertices, big if it has between 24 and 41 vertices, and monster if it has more than 41 vertices.

6.1

Medium Faces

Prior to Nicholson and Sneddon improving the lower bound on the order of a PCC graph to 208, the largest known PCC graph was a 138-vertex graph in PCC∗_{4,5,19} due to Reti, Bitay, and Kosztolanyi. It has six 19-faces. In the same paper, Reti, Bitay, and Kosztolanyi give the PCC graphs G20 and G22 depicted in Figure 6.1, which have respectively two 20-faces and two 22-faces.

Figure 6.1: G20 and G22

A class of PCC graphs can be constructed by using the two motifs pictured in Figure 6.2. Edges are added with green vertices as their endpoints to form the exterior face and with red vertices as their endpoints to form the interior face. For example, see Figure 6.3.

Figure 6.2: Two motifs X and Y

For interior/exterior faces of size k, the construction yields four vertices of type {3, 8, k} for each motif. This constrains k to be at most 23. As a result, whenever the inequalities 3 ≤ 3x + 2y ≤ k ≤ 23 are satisfied by non-negative integers x, y, and k, a PCC graph can be constructed with two k-faces and x + y octagons using x copies of X and y copies of Y . For example, we can maximize k by setting x = 1 and y = 10, which gives

us the PCC graph pictured in 6.3 with two faces of size 23, which we call

G23. The graph G23 is a counterexample to Conjectures 6.1 and 6.2, since it has faces of size 23 and vertices of type{3, 8, 23} which have curvature ₅₅₂1 .

x y # graphs 0 10 1 2 7 4 4 4 8 6 1 1 Total 14 x y # graphs 1 9 1 3 6 7 5 3 5 7 0 1 Total 14 k = 22 x y # graphs 0 11 1 2 8 5 4 5 10 6 2 4 Total 20 k = 23 x y # graphs 1 10 1 3 7 8 5 4 10 7 1 1 Total 20

Figure 6.4: The number of PCC graphs constructed with two k-faces

For each solution to the inequality, the motifs X and Y can be arranged to create as many distinct graphs as the number of bracelets with x red and

y white beads. The numerical solutions to the inequalities along with the

number of corresponding graphs created by the construction for k equal to 20, 21, 22, or 23 are given in Figure 6.4. The graphs corresponding to the solutions for k = 20, including G22from Figure 6.1, are shown in Figure 6.5.

For k = 20 and k = 21 there are 14 graphs each, while for k = 22 and

of 14 + 14 + 20 + 20 = 68 graphs with faces of size 20 or greater, whereas Conjecture 6.1 states that there are just two.

6.2

Big Faces

A class of PCC graphs can be constructed from between one and thirteen copies of the motif in Figure 6.6 by using the coloured vertices as endpoints for the edges with the same colours. The resulting graph has 16x vertices and two faces of size 3x where x is the number of copies used. When x = 13 the constructed graph has 208 vertices. This graph is shown in Figure 6.7. Nicholson and Sneddon found two other PCC graphs with 208 vertices and no bigger ones are known.

Figure 6.6: A motif for constructing PC graphs with large faces.

The construction can be used to construct a graph with y-faces for any y that is a multiple of three and at most 39. It is an open problem to deter-mine if there are any PCC graphs with a face of size greater than 23 and not a multiple 3.

Figure 6.8 gives a projective planar analogue with one 39-face and 104 ver-tices.

6.3

Monster Faces

Theorem 6.3 A PCC graph has at most one monster face (face of size at least 42).

Proof. A monster face B of a PCC graph G of size b must consist entirely of vertices of type{4, 4, b}, {3, 3, 3, b}, or {3, x, b} with 3 ≤ x ≤ 6 (See Table 2.1). Each of these vertex types has curvature at least 1_b, so the sum of the curvatures of all the vertices of B is at least 1.

If B it consists entirely of vertices of type {4, 4, b} or entirely of vertices of type {3, 3, 3, b}, then G is a prism or antiprism. If it has both vertices of type{4, 4, b} and vertices of type {3, 3, 3, b}, then it must also have at least two vertices with type {3, 4, b}. Thus for a PCC graph to have a monster face of size b, it must have at least one vertex u of the form {3, y, b} with 3≤ y ≤ 6. Vertex u must have a neighbour v that is not on B, and, since G is a PCC graph, v must have strictly positive curvature. Since the vertices of B have curvatures summing to at least one and v has positive curvature, the sum of the curvatures of B∪ {v} is strictly greater than one, hence a PCC graph can have at most one monster face. □

CHAPTER 7

Bounding PCC Subclasses

In this section some interesting subclasses of PCC are discussed. A program with a heuristic approach was used to search for 3-regular PCC graphs given the number of vertices n and a set S of face sizes. Restricting the ILP solutions to PCC∗S is done by adding the constraint:

∑

s∈S

s = f . (7.1)

Nicholson and Sneddon [7] suggested that the vertex types {4, 7, 9} and

{3, 10, 14} might help to find a PCC graph with more than 208 vertices. In

this section we find tight bounds for PCC∗_{4,7,9} and PCC_{3,10,14}.

For PCC∗_{4,7,9} SCIP gives the ILP solution in Table 7.1. This is realized by the graph in Figure 7.1.

Variable Value

n 62

x(4,4,7) 8 x(4,4,9) 6 x(4,7,9) 48

Table 7.1: PCC∗_{4,7,9} ILP solutions

Figure 7.1: A maximum order PCC∗_{4,7,9} graph

For PCC∗_{3,10,14} SCIP gives the ILP solution in Table 7.2 (a). Constraint 5.3, which was used specifically for the 3-regular case is also valid in general and after adding it to this case SCIP gives the solution in Table 7.2 (b), with a bound of 108. The largest graph that the heuristic program found has 84 vertices and is shown in Figure 7.2.

Variable Value n 132 f3 44 f10 18 f14 6 x(3,10,10) 48 x(3,10,14) 84 (a) Variable Value n 108 f3 36 f10 16 f14 4 x(3,10,10) 52 x(3,10,14) 56 (b) Table 7.2: PCC∗_{3,10,14} ILP solutions

CHAPTER 8

Previous Work on Bounding the General Case

DeVos and Mohar [3] showed that a PCC graph cannot have more than 3444 vertices by using a discharging argument based on the fact that a vertex v, of a PCC graph, with curvature less than ₁₇₂₂1 must have some neighbour with curvature greater than ₁₇₂₂1 that can can send some curvature as charge to v.

Zhang [6] claimed to lower the upper bound to 580, but the proof relies on the following lemma:

Lemma 8.1 Let G be a finite simple graph, with at least 290 vertices, em-bedded in a projective space such that every vertex has positive curvature and degree at least 3, then all vertices of G are on a cycle bounding a face of length |V (G)|.

The proof Zhang gave for Lemma 8.1 is not correct. It considers what hap-pens if every vertex has curvature less than ₂₉₀1 or every vertex has curvature at least ₂₉₀1 , but does not consider the case where there are some vertices with curvature less than ₂₉₀1 and others with curvature at least ₂₉₀1 , which is certainly possible. For instance the graph in Figure 6.8 is a PCC graph embedded in the projective plane which has vertices of type {3, 7, 39} which have curvature less than ₂₉₀1 and vertices of types{3, 3, 3, 39} and {3, 3, 5, 7} which both have curvature greater than ₂₉₀1 .

Since Zhang’s proof is not correct the upper bound of 580 is not established and the best known upper bound is 3444.

CHAPTER 9

Bounding the General Case

In this chapter the ILP solutions are used to provide insight towards improv-ing the general upper bound on the order of PCC graphs. If Conjecture 6.4 (which states that there are no PCC graphs with faces of size 42 or greater) is true, then by Theorem 9.1, 244 is a general upper bound.

Theorem 9.1 If a PCC graph has no faces with size larger than 122, then it has maximum order 244.

Proof. The ILP solutions found in Chapter 3 give an upper bound on the order of a PCC∗_{{3,4,...,132}} graph of 264, as per Table 3.3. The values of an optimal solution to the ILP found by SCIP, are given in Table 9.1. A PCC graph realizing the ILP solution has six 4-cycles with cycle sequence type

y3,3,3,3 and twelve with type y3,11,3,11, which correspond to the motifs Type A and Type C in Figure 9.1.

Type A Type B Type C

Variable Value Vertex Sequence Types

x(4,6,11) 192 x(3,3,4,11) 48 x(3,4,3,11) 24

n 264

3-face Cycle Sequence Types

y3,4,11 48

4-face Cycle Sequence Types

y3,3,3,3 6

y3,11,3,11 12 y6,11,6,11 48 6-face Cycle Sequence Types

y4,11,4,11,4,11 32 11-face Cycle Sequence Types

y3,3,4,6,4,6,4,6,4,6,4 24

Table 9.1: General ILP solution

3, [3], 11, [3], 3, [3], 11, [3] 3, [3, 3], 3, [3, 3], 3, [3, 3], 3, [3, 3] Figure 9.2: 4-faces with augmented cycle types

augmented to include the face sizes of faces that are incident to vertices but not edges of the face, in clockwise order. For example, the central face of the graph on the left of Figure 9.2 has cycle sequence type 3, 11, 3, 11 and

augmented cycle sequence type 3, [3], 11, [3], 3, [3], 11, [3] (the extra cycle sizes

are given in [ ]’s) and the central face of the graph on the right has cycle sequence type 3, 3, 3, 3 and augmented cycle sequence type 3, [3, 3], 3, [3, 3], 3, [3, 3], 3, [3, 3]. Each 4-face motif has a distinct augmented cycle sequence type. For each augmented cycle sequence type r, a variable zris added to the

ILP, representing the number of faces in the solution that have augmented cycle sequence type r.

The vertex sequence types of the vertices on the 4-face can be determined from the augmented cycle type. For example an augmented cycle sequence type of the form q, [], r, [s], t, [u, v], w, [x] has vertices with vertex sequence types (4, q, r), (4, r, s, t), (4, t, u, v, w), and (4, w, x, q). The central face of the graph on the left of Figure 9.2 has two vertices of vertex sequence type (4, 3, 3, 11) and two vertices of type (4, 11, 3, 3), which both have canonical form (3, 3, 4, 11). The central face of the graph on the right of Figure 9.2 has four vertices with vertex sequence types (3, 3, 3, 3, 4) in canonical form.

The vertex sequence types are counted by augmented cycle type. Let A be the set of all augmented cycle sequence types and let m(t, r) denote the number of types that the vertex sequence type t occurs in the augmented cycle type r. For each vertex sequence type t such that 4∈ t a constraint is added:

m(4, t) xt=

∑

r∈A

Notice that if an augmented cycle sequence type has [11], 3, [11] as a subse-quence, then the 3-face that shares an edge with the 4-face has cycle sequence type 4, 11, 11. For each augmented cycle sequence type r there must be f (r) 3-faces with cycle sequence type 4, 11, 11 where f (r) is the number of occur-rences of [11], 3, [11] in r. For example, Type A motifs have four occuroccur-rences of the subsequence [11], 3, [11], while Type B motifs have one and Type C have none. The constraint added is:

∑

r∈A

f (r) zr ≤ y4,11,11 (9.2)

Similarly, for every augmented cycle sequence type r with the subsequence [3], 3, [3], there must be g(r) 3-faces with cycle sequence type 3, 3, 4 where

g(r) is the number of occurrences of [3], 3, [3] in r. Type A and B motifs do

not have such a subsequence, but Type C motifs have two occurrences of it. The constraint added is:

∑

r∈A

g(r) zr ≤ y3,3,4 (9.3)

Adding constraints 9.1, 9.2, and 9.3 to the ILP and solving with SCIP again gives the solution in Table 9.2, which has n = 244 and is the tightest bound we have under the assumption that are no monster faces. □

Variable Value Vertex Sequence Types

x(3,3,4,11) 40

x(3,4,3,11) 40

x(3,11,11) 20

x(4,6,11) 144

n 244

3-face Cycle Sequence Types

y3,4,11 40

y4,11,11 20

4-face Augmented Cycle Sequence Types

z3,[11],3,[11],3,[3],11,[3] 20 z6,[],11,[],6,[],11,[] 36 6-face Cycle Sequence Types

y4,11,4,11,4,11 24 11-face Cycle Sequence Types

y3,3,3,3,3,3,3,3,3,3,4 2 y3,3,4,6,4,6,4,6,4,6,4 18 y3,3,11,3,11,3,11,3,11,3,11 4

CHAPTER 10

Conclusions

The previously shown lower bounds for the number of vertices of a 3-regular PCC is 132 and for a general PCC is 208. In this thesis it was shown that the 3-regular lower bound is tight. Using integer linear programming and several theoretical results, we proved that there are five 3-regular PCC graphs of maximum order 132. For the general case, a new polyhedron realizing the current best lower bound of 208 was constructed.

Future research on improving the lower bound could involve better tactics for generating PCC graphs that correspond to a solution to the ILP. Adding various constraints to the ILP might reveal solutions can give solutions that correspond to actual PCC graphs.

The most interesting open questions are:

• Are there any PCC graphs with more than 208 vertices?

• Are there any PCC graphs with faces of size 40 or more or any of size

greater than 23 where n is not divisible by three?

The tactics applied here have also been shown to work well for defining the maximum size for a PCC graph given a restricted set of face sizes.

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All Valid Vertex Types with Face Sizes at Most 41

This table includes all vertex types t which have positive combinatorial cur-vature and are incident to faces of size 41 or less. Beside each vertex type

t is the value

⌊ 2 Φ(t)

⌋

which is a trivial upper bound on the order of a PCC graph using only vertices of type t or vertices that come after it in the table.

{3,7,41} 3444 {3,3,4,11} 264 {4,6,10} 120 {4,4,39} 78 {3,11,13} 1716 {3,8,20} 240 {4,5,15} 120 {3,3,3,38} 76 {3,7,40} 1680 {3,7,31} 236 {3,8,17} 116 {4,4,38} 76 {3,8,23} 1104 {4,5,17} 226 {4,7,8} 112 {3,7,20} 76 {3,7,39} 1092 {3,7,30} 210 {3,7,24} 112 {3,6,38} 76 {3,7,38} 798 {3,3,5,7} 210 {3,7,23} 101 {4,5,13} 74 {4,5,19} 760 {5,6,7} 210 {3,8,16} 96 {3,3,3,37} 74 {3,7,37} 621 {3,10,13} 195 {4,5,14} 93 {3,6,37} 74 {3,9,17} 612 {3,7,29} 187 {3,9,13} 93 {4,4,37} 74 {3,8,22} 528 {3,8,19} 182 {3,7,22} 92 {3,6,36} 72 {3,7,36} 504 {3,9,15} 180 {3,7,21} 84 {3,9,12} 72 {4,7,9} 504 {5,5,9} 180 {3,10,11} 82 {4,6,9} 72 {3,7,35} 420 {3,7,28} 168 {3,3,3,41} 82 {4,4,36} 72 {3,10,14} 420 {4,5,16} 160 {3,6,41} 82 {3,3,3,36} 72 {4,5,18} 360 {3,7,27} 151 {4,4,41} 82 {3,3,4,9} 72 {3,7,34} 357 {3,8,18} 144 {4,4,40} 80 {3,3,3,35} 70 {3,8,21} 336 {3,7,26} 136 {3,6,40} 80 {4,4,35} 70 {3,7,33} 308 {3,11,11} 132 {3,3,3,40} 80 {3,6,35} 70 {3,9,16} 288 {3,9,14} 126 {3,8,15} 80 {3,7,19} 69 {3,7,32} 268 {3,7,25} 123 {5,5,8} 80 {4,4,34} 68 {3,11,12} 264 {3,10,12} 120 {3,3,3,39} 78 {3,6,34} 68 {4,6,11} 264 {3,3,4,10} 120 {3,6,39} 78 {3,3,3,34} 68

{3,8,14} 67 {4,4,27} 54 {3,3,3,20} 40 {3,5,30} 30 {3,6,33} 66 {3,6,26} 52 {3,3,3,19} 38 {3,5,32} 30 {3,3,3,33} 66 {3,3,3,26} 52 {4,4,19} 38 {5,5,6} 30 {4,4,33} 66 {4,4,26} 52 {3,6,19} 38 {3,6,15} 30 {3,3,3,32} 64 {3,7,16} 51 {3,7,13} 37 {3,3,3,15} 30 {4,4,32} 64 {3,6,25} 50 {4,4,18} 36 {3,5,29} 29 {3,6,32} 64 {3,3,3,25} 50 {3,3,3,18} 36 {3,7,11} 29 {3,7,18} 63 {4,4,25} 50 {3,9,9} 36 {3,3,3,14} 28 {4,4,31} 62 {3,3,3,24} 48 {3,6,18} 36 {4,4,14} 28 {3,6,31} 62 {4,5,11} 48 {4,4,17} 34 {3,6,14} 28 {3,3,3,31} 62 {4,4,24} 48 {3,8,10} 34 {3,5,27} 28 {3,4,4,5} 60 {3,8,12} 48 {3,6,17} 34 {3,5,28} 28 {3,10,10} 60 {4,6,8} 48 {3,5,40} 34 {3,8,9} 28 {3,3,3,3,5} 60 {3,3,4,8} 48 {3,5,41} 34 {3,5,25} 27 {3,6,30} 60 {3,6,24} 48 {3,3,3,17} 34 {3,5,26} 27 {5,6,6} 60 {3,3,3,23} 46 {3,5,38} 33 {3,5,23} 26 {3,3,5,6} 60 {3,6,23} 46 {3,5,37} 33 {3,6,13} 26 {3,3,3,30} 60 {4,4,23} 46 {4,6,7} 33 {3,5,24} 26 {4,4,30} 60 {3,7,15} 46 {3,3,4,7} 33 {4,4,13} 26 {4,5,12} 60 {5,5,7} 46 {3,7,12} 33 {3,7,10} 26 {4,4,29} 58 {3,9,10} 45 {3,5,39} 33 {3,3,3,13} 26 {3,6,29} 58 {3,3,3,22} 44 {3,5,35} 32 {4,5,8} 26 {3,3,3,29} 58 {4,4,22} 44 {3,3,3,16} 32 {3,5,22} 25 {3,7,17} 57 {3,6,22} 44 {3,6,16} 32 {3,8,8} 24 {4,7,7} 56 {3,7,14} 42 {4,4,16} 32 {3,5,21} 24 {3,6,28} 56 {3,3,3,21} 42 {4,5,9} 32 {3,4,4,4} 24 {3,8,13} 56 {3,6,21} 42 {3,5,36} 32 {4,4,12} 24 {4,4,28} 56 {4,4,21} 42 {3,5,34} 31 {3,5,20} 24 {3,9,11} 56 {4,4,20} 40 {3,5,33} 31 {3,3,4,6} 24 {3,3,3,28} 56 {4,5,10} 40 {4,4,15} 30 {3,3,3,3,4} 24 {3,3,3,27} 54 {3,8,11} 40 {3,5,31} 30 {3,6,12} 24 {3,6,27} 54 {3,6,20} 40 {3,3,5,5} 30 {4,6,6} 24

{3,5,19} 23 {3,4,34} 17 {3,4,14} 12 {3,3,21} 9 {3,3,3,11} 22 {3,4,35} 17 {3,3,3,6} 12 {3,3,20} 9 {3,7,9} 22 {3,5,12} 17 {3,4,12} 12 {3,3,19} 9 {3,6,11} 22 {3,4,29} 16 {4,4,6} 12 {3,3,18} 9 {3,5,18} 22 {3,5,11} 16 {3,3,4,4} 12 {3,3,29} 9 {4,4,11} 22 {3,3,3,8} 16 {3,3,3,3,3} 12 {3,3,22} 9 {3,5,17} 21 {4,4,8} 16 {3,4,11} 11 {3,4,6} 8 {4,5,7} 21 {3,6,8} 16 {3,5,7} 11 {3,3,16} 8 {3,3,3,10} 20 {3,4,24} 16 {3,3,37} 10 {3,3,15} 8 {4,4,10} 20 {3,4,25} 16 {4,4,5} 10 {3,3,14} 8 {3,6,10} 20 {3,4,26} 16 {3,5,6} 10 {3,3,13} 8 {3,5,16} 20 {3,7,7} 16 {3,3,35} 10 {3,3,12} 8 {3,5,15} 20 {3,4,27} 16 {3,3,30} 10 {3,5,5} 8 {5,5,5} 20 {3,4,28} 16 {3,3,41} 10 {4,4,4} 8 {3,5,14} 19 {3,5,10} 15 {3,3,40} 10 {3,3,3,4} 8 {3,7,8} 19 {3,4,20} 15 {3,3,39} 10 {3,4,7} 8 {3,4,38} 18 {3,4,23} 15 {3,3,38} 10 {3,3,17} 8 {3,4,37} 18 {3,4,21} 15 {3,4,10} 10 {3,3,9} 7 {4,4,9} 18 {3,4,22} 15 {3,3,36} 10 {3,3,10} 7 {3,3,3,9} 18 {3,4,19} 14 {3,4,9} 10 {3,3,11} 7 {3,4,36} 18 {3,4,18} 14 {3,3,34} 10 {3,4,5} 7 {3,5,13} 18 {3,3,3,7} 14 {3,3,33} 10 {3,4,4} 6 {3,4,41} 18 {3,6,7} 14 {3,3,32} 10 {3,3,8} 6 {3,4,40} 18 {4,4,7} 14 {3,3,31} 10 {3,3,7} 6 {3,4,39} 18 {3,4,17} 14 {3,3,3,5} 10 {3,3,6} 6 {3,6,9} 18 {3,5,9} 13 {3,4,8} 9 {3,3,3,3} 6 {3,4,30} 17 {3,4,16} 13 {3,3,27} 9 {3,3,5} 5 {3,4,31} 17 {3,4,15} 13 {3,3,26} 9 {3,3,4} 4 {3,4,32} 17 {4,5,5} 13 {3,3,25} 9 {3,3,3} 4 {3,4,33} 17 {3,4,13} 12 {3,3,24} 9 {4,5,6} 17 {3,6,6} 12 {3,3,23} 9