Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with - - PowerPoint PPT Presentation

Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed Automorphism utomorphism Group Group

C.J. Jones, Sarah E. Lubow, and Carlie J. Triplitt

University of Texas at Tyler

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Automorphism Group of a Graph

Definition

A permutation of a set S is a bijection from S to itself.

Definition

Let Γ be a graph. The automorphism group of Γ, denoted Aut Γ, is the set of adjacency preserving permutations of the vertices of Γ.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Example

Note that Z3 ∼ = {1, 2, 3} ∼ = (1, 2, 3)(4, 5, 6)(7, 8, 9). In this case, we have that Aut Γ = {(1), (1, 2, 3)(4, 5, 6)(7, 8, 9), (1, 3, 2)(4, 6, 5)(7, 9, 8)}. 7 8 9 4 5 6 1 2 3

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Connected Components

Definition

A connected component is a subgraph in which any two vertices are connected to each other by paths, and that is connected to no additional vertices in the supergraph. 1 3 2 4 6 5

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Planar Graphs

Definition

A graph is planar if it can be draw so that no edges intersect. 3 4 1 2 3 4 1 2 K4 graph and a planar embedding.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

F ∗-diagrams

We can use F ∗-diagrams to depict graphs that are too complicated to draw explicitly.

U3(1) V3 W3 X2(1)

0, 1

(a) F ∗-diagram of Γ

u1 u2 u3 v1 v2 v3 w1 w2 w3 x1 x2

(b) Depiction of Γ January 27, 2019 NCUWM Presentation 6/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

α(G)

Definition

For a finite group G, let α(G) denote the minimum number of vertices among all graphs Γ such that Aut Γ ∼ = G.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Value of α(G)

Theorem

The value of α(G) has been established for the following groups G: finite abelian groups (Arlinghaus); hyperoctahedral groups (Haggard, McCarthy, Wohlgemuth); symmetric groups (Quintas); alternating groups of degree at least 13 (Liebeck); generalized quaternion groups (Graves, Graves, Lauderdale); and dihedral groups (Graves, Graves, Haggard, Lauderdale, McCarthy).

January 27, 2019 NCUWM Presentation 8/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Vertex-Minimal Planar Graphs

Maruˇ siˇ c was the first to consider vertex-minimal planar graphs with a prescribed automorphism group.

Definition

For a finite group G, we let αP(G) denote the minimum number of vertices among all planar graphs Γ such that Aut Γ ∼ = G. If no planar graph Γ satisfies Aut Γ ∼ = G, then we define αP(G) = ∞.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Vertex-Minimal Planar Graphs

Maruˇ siˇ c was the first to consider vertex-minimal planar graphs with a prescribed automorphism group.

Definition

For a finite group G, we let αP(G) denote the minimum number of vertices among all planar graphs Γ such that Aut Γ ∼ = G. If no planar graph Γ satisfies Aut Γ ∼ = G, then we define αP(G) = ∞. It is clear that α(G) ≤ αP(G) for all finite groups G.

January 27, 2019 NCUWM Presentation 9/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Vertex-Minimal Planar Graphs

Let Zm denote the cyclic group of order m.

Theorem (Maruˇ siˇ c, 1980)

Assume m = pa1

1 · pa2 2 · . . . · pak k

is the prime factorization of the integer m, where ai ≥ 1 for all i ∈ [k]. If m is odd, then αP(Zm) = 3(pa1

1 + pa2 2 + . . . + pak k ).

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Vertex-Minimal Planar Graphs

Maruˇ siˇ c conjectured that a similar result holds when m = 2n. Archer et al. proved Maruˇ siˇ c’s conjecture with the following theorem.

Theorem (Archer, Darby, Lauderdale, Linson, Maxfield, Schmidt, Tran, 2017)

If m = 2s with s ≥ 1, then αP(Zm) =

• 2

if s = 1 2m + 2 if s > 1.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Vertex-Minimal Planar Graphs

Maruˇ siˇ c conjectured a similar result for when m is an even and not a power of 2. We proved Maruˇ siˇ c’s conjecture, which is stated in the following theorem.

Theorem (Jones, L., T., 2018)

If m = 2s · pa1

1 · pa2 2 · . . . · pak k

is the prime factorization of the integer m, where ai ≥ 1 for all i ∈ [k] and k ≥ 1, then αP (Zm) =

• 2 + 3(pa1

1 + pa2 2 + . . . + pak k )

if s = 1 2 + 2s+1 + 3(pa1

1 + pa2 2 + . . . + pak k )

if s > 1.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Example of Theorem:

Our results prove that αP (Z1008) = 82. Since 1008 = 24 · 32 · 7, then Γ1,008 = Γ′

16 + Γ′ 9 + Γ′ 7.

U7(1) V7 W7 X9(1) Y9 Z9 A16(1) B16 C2

0, 1 0, 1 0, 1

F ∗-diagram of Γ1,008

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Connected Cyclic Graphs

Maruˇ siˇ c conjectured that the connected cyclic vertex-minimal graph would have just one additional vertex.

Theorem (Jones, L., T., 2018)

If m = 2s · pa1

1 · pa2 2 · . . . · pak k

is the prime factorization of the integer m, where ai ≥ 1 for all i ∈ [k] and k ≥ 1, then αP

c (Zm) =

• 3(pa1

1 + pa2 2 + . . . + pak k ) + 3

if s = 1 3(2s + pa1

1 + pa2 2 + . . . + pak k ) + 1

if s > 1.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Example αP

c (Z12)

Our results prove that αP

c (Z12) = 22. U3(1) V3 W3 X4(1) Y4 Z4 A1

0, 1 0, 1

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Definition

A dihedral group has presentation D2n = r, s : rn = 1 = s2, rs = sr−1, and is often thought of as the symmetries of a regular n-gon. The value of αP (D2n) denotes the order of a vertex-minimal planar graph with dihedral symmetry.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

We know the lower bound of αP (D2n) is α(D2n), and the values of α(D2n) were found by Haggard, McCarthy, Graves, Graves and Lauderdale. Additionally we know the upper bound corresponds to the

• rder of an n-gon. An n-gon is planar.

January 27, 2019 NCUWM Presentation 17/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Let’s look at an example where α(D2n) = αP (D2n). Assume n = 136, 191, 250.

January 27, 2019 NCUWM Presentation 18/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Let’s look at an example where α(D2n) = αP (D2n). Assume n = 136, 191, 250. In this case, α(D2n) = 1, 044, and αP (D2n) = 68, 095, 625 + 2.

January 27, 2019 NCUWM Presentation 18/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Lemma

Let D2n be a dihedral group, where n is twice an odd

• number. Then D2n ∼

= Dn × Z2. Otherwise D2n is directly indecomposable. For n = 136, 191, 250 we now have two components, representing D68,095,625 and Z2. U68,095,625(1) V2(1)

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Theorem (Jones, L., T., 2018)

Let D2n be a dihedral group for some n ≥ 3. If n is twice an odd integer, then αP (D2n) = n

2 + 2. Otherwise,

αP (D2n) = n. Notice n = 136, 191, 250 falls into the special case.

January 27, 2019 NCUWM Presentation 20/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Dihedral Groups

Theorem (Jones, L., T., 2018)

Let D2n be a dihedral group for some n ≥ 3. If n is twice an odd integer, then αP (D2n) = n

2 + 2. Otherwise,

αP (D2n) = n. Notice n = 136, 191, 250 falls into the special case.Thus proving the order of a vertex-minimal planar graph with dihedral symmetry.

January 27, 2019 NCUWM Presentation 20/24

Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Future Work

Theorem (Jones, L., T., 2018)

Let p be a prime number.

• 1. If p = 2, then αP (Zp × Zp) = 4.
• 2. If p > 2, then αP (Zp × Zp) = 6p.

Open Question.

What is the value of αP (G), when G is a finite noncyclic abelian group?

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Future Work

Conjecture (Jones, L., T., 2018).

Assume that G ∼ = Zp × Zp × ... × Zp, where n is the number of times Zp appears and p is an odd prime. If n is even, then αP (G) = 2

n 2

• i=1

(i + 2)p and if n is odd, then αP (G) = n + 5 2

• p + 2

n−1 2

• i=1

(i + 2)p.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Acknowledgements

This research was supported by the National Science Foundation (DMS-1659221). Additionally, we would like to thank the University of Texas at Tyler, especially Lindsey-Kay Lauderdale, David Milan, and Christina Graves, for their support in conducting this research.

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Introduction Background F ∗-diagrams α(G) and αP (G) αP (G) Conjectures and Theorems αP

c (Zm)

Dihedral Groups Future Work

Gratitude

Thank You! colin.jones@eastern.edu selubow@my.loyno.edu ctri8247@usao.edu

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