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

Automorphism Group of a Graph Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) α P ( G ) Definition Conjectures A permutation of a set S is a bijection from S to itself. and Theorems α P c ( Z m ) Dihedral Definition Groups Let Γ be a graph. The automorphism group of Γ, Future Work denoted Aut Γ, is the set of adjacency preserving permutations of the vertices of Γ. January 27, 2019 NCUWM Presentation 2/24

Example Introduction Note that Z 3 ∼ = { 1 , 2 , 3 } ∼ Background = � (1 , 2 , 3)(4 , 5 , 6)(7 , 8 , 9) � . In this F ∗ -diagrams α ( G ) and case, we have that α P ( G ) α P ( G ) Aut Γ = { (1) , (1 , 2 , 3)(4 , 5 , 6)(7 , 8 , 9) , (1 , 3 , 2)(4 , 6 , 5)(7 , 9 , 8) } . Conjectures and Theorems 7 α P c ( Z m ) Dihedral 4 Groups Future Work 2 5 1 3 8 9 6 January 27, 2019 NCUWM Presentation 3/24

Connected Components Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) Definition α P ( G ) A connected component is a subgraph in which any two Conjectures vertices are connected to each other by paths, and that is and Theorems α P connected to no additional vertices in the supergraph. c ( Z m ) Dihedral Groups 2 5 Future Work 1 3 4 6 January 27, 2019 NCUWM Presentation 4/24

Planar Graphs Introduction Background Definition F ∗ -diagrams α ( G ) and A graph is planar if it can be draw so that no edges α P ( G ) intersect. α P ( G ) Conjectures and Theorems α P c ( Z m ) Dihedral Groups 1 2 1 2 Future Work 3 4 3 4 K 4 graph and a planar embedding. January 27, 2019 NCUWM Presentation 5/24

F ∗ -diagrams Introduction Background F ∗ -diagrams α ( G ) and We can use F ∗ -diagrams to depict graphs that are too α P ( G ) complicated to draw explicitly. α P ( G ) Conjectures and Theorems x 2 x 1 w 1 α P c ( Z m ) 0 , 1 U 3 (1) X 2 (1) u 1 v 2 V 3 Dihedral v 1 Groups u 3 Future Work u 2 W 3 w 3 v 3 w 2 (a) F ∗ -diagram of Γ (b) Depiction of Γ January 27, 2019 NCUWM Presentation 6/24

α ( G ) Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) α P ( G ) Conjectures Definition and Theorems α P c ( Z m ) For a finite group G , let α ( G ) denote the minimum Dihedral number of vertices among all graphs Γ such that Aut Groups Γ ∼ = G . Future Work January 27, 2019 NCUWM Presentation 7/24

Value of α ( G ) Introduction Theorem Background F ∗ -diagrams The value of α ( G ) has been established for the following α ( G ) and α P ( G ) groups G : α P ( G ) finite abelian groups ( Arlinghaus ); Conjectures and Theorems hyperoctahedral groups ( Haggard, McCarthy, α P c ( Z m ) Wohlgemuth ); Dihedral Groups symmetric groups ( Quintas ); Future Work 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

Vertex-Minimal Planar Graphs Introduction Background F ∗ -diagrams Maruˇ siˇ c was the first to consider vertex-minimal planar α ( G ) and α P ( G ) graphs with a prescribed automorphism group. α P ( G ) Conjectures and Theorems Definition α P c ( Z m ) For a finite group G , we let α P ( G ) denote the minimum Dihedral Groups number of vertices among all planar graphs Γ such that Future Work Aut Γ ∼ = G . If no planar graph Γ satisfies Aut Γ ∼ = G , then we define α P ( G ) = ∞ . January 27, 2019 NCUWM Presentation 9/24

Vertex-Minimal Planar Graphs Introduction Background F ∗ -diagrams Maruˇ siˇ c was the first to consider vertex-minimal planar α ( G ) and α P ( G ) graphs with a prescribed automorphism group. α P ( G ) Conjectures and Theorems Definition α P c ( Z m ) For a finite group G , we let α P ( G ) denote the minimum Dihedral Groups number of vertices among all planar graphs Γ such that Future Work Aut Γ ∼ = G . If no planar graph Γ satisfies Aut Γ ∼ = G , then we define α P ( G ) = ∞ . It is clear that α ( G ) ≤ α P ( G ) for all finite groups G . January 27, 2019 NCUWM Presentation 9/24

Vertex-Minimal Planar Graphs Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) Let Z m denote the cyclic group of order m . α P ( G ) Conjectures and Theorems Theorem (Maruˇ siˇ c, 1980) α P c ( Z m ) 2 · . . . · p a k Assume m = p a 1 1 · p a 2 Dihedral is the prime factorization of Groups k the integer m , where a i ≥ 1 for all i ∈ [ k ] . If m is odd, Future Work then α P ( Z m ) = 3( p a 1 1 + p a 2 2 + . . . + p a k k ) . January 27, 2019 NCUWM Presentation 10/24

Vertex-Minimal Planar Graphs Introduction Background F ∗ -diagrams Maruˇ siˇ c conjectured that a similar result holds when α ( G ) and m = 2 n . Archer et al. proved Maruˇ α P ( G ) siˇ c’s conjecture with α P ( G ) the following theorem. Conjectures and Theorems α P c ( Z m ) Theorem (Archer, Darby, Lauderdale, Linson, Dihedral Maxfield, Schmidt, Tran, 2017) Groups Future Work If m = 2 s with s ≥ 1 , then � 2 if s = 1 α P ( Z m ) = 2 m + 2 if s > 1 . January 27, 2019 NCUWM Presentation 11/24

Vertex-Minimal Planar Graphs Introduction Background F ∗ -diagrams Maruˇ siˇ c conjectured a similar result for when m is an even α ( G ) and α P ( G ) and not a power of 2. We proved Maruˇ siˇ c’s conjecture, α P ( G ) which is stated in the following theorem. Conjectures and Theorems Theorem (Jones, L., T., 2018) α P c ( Z m ) If m = 2 s · p a 1 1 · p a 2 2 · . . . · p a k Dihedral is the prime factorization of Groups k the integer m , where a i ≥ 1 for all i ∈ [ k ] and k ≥ 1 , then Future Work � 2 + 3( p a 1 1 + p a 2 2 + . . . + p a k k ) if s = 1 α P ( Z m ) = 2 + 2 s +1 + 3( p a 1 1 + p a 2 2 + . . . + p a k k ) if s > 1 . January 27, 2019 NCUWM Presentation 12/24

Example of Theorem: Introduction Background Our results prove that α P ( Z 1008 ) = 82 . F ∗ -diagrams α ( G ) and Since 1008 = 2 4 · 3 2 · 7, then Γ 1 , 008 = Γ ′ α P ( G ) 16 + Γ ′ 9 + Γ ′ 7 . α P ( G ) 0 , 1 0 , 1 Conjectures U 7 (1) V 7 A 16 (1) B 16 and Theorems α P c ( Z m ) Dihedral W 7 C 2 Groups 0 , 1 X 9 (1) Y 9 Future Work Z 9 F ∗ -diagram of Γ 1 , 008 January 27, 2019 NCUWM Presentation 13/24

Connected Cyclic Graphs Introduction Background F ∗ -diagrams Maruˇ siˇ c conjectured that the connected cyclic α ( G ) and α P ( G ) vertex-minimal graph would have just one additional α P ( G ) vertex. Conjectures and Theorems Theorem (Jones, L., T., 2018) α P c ( Z m ) If m = 2 s · p a 1 1 · p a 2 2 · . . . · p a k Dihedral is the prime factorization of k Groups the integer m , where a i ≥ 1 for all i ∈ [ k ] and k ≥ 1 , then Future Work 2 + . . . + p a k � 3( p a 1 1 + p a 2 k ) + 3 if s = 1 α P c ( Z m ) = 3(2 s + p a 1 1 + p a 2 2 + . . . + p a k k ) + 1 if s > 1 . January 27, 2019 NCUWM Presentation 14/24

Example α P c ( Z 12 ) Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) α P ( G ) Our results prove that α P c ( Z 12 ) = 22 . Conjectures and Theorems α P c ( Z m ) 0 , 1 0 , 1 U 3 (1) X 4 (1) V 3 Y 4 Dihedral Groups A 1 Future Work W 3 Z 4 January 27, 2019 NCUWM Presentation 15/24

Dihedral Groups Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) Definition α P ( G ) A dihedral group has presentation Conjectures and Theorems D 2 n = � r, s : r n = 1 = s 2 , rs = sr − 1 � , α P c ( Z m ) Dihedral Groups and is often thought of as the symmetries of a regular Future Work n -gon. The value of α P ( D 2 n ) denotes the order of a vertex-minimal planar graph with dihedral symmetry. January 27, 2019 NCUWM Presentation 16/24

Dihedral Groups Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) α P ( G ) We know the lower bound of α P ( D 2 n ) is α ( D 2 n ), and the Conjectures and Theorems values of α ( D 2 n ) were found by Haggard, McCarthy, α P c ( Z m ) Graves, Graves and Lauderdale. Dihedral Groups Additionally we know the upper bound corresponds to the Future Work order of an n -gon. An n -gon is planar. January 27, 2019 NCUWM Presentation 17/24

Dihedral Groups Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) Let’s look at an example where α ( D 2 n ) � = α P ( D 2 n ). α P ( G ) Conjectures Assume n = 136 , 191 , 250. and Theorems α P c ( Z m ) Dihedral Groups Future Work January 27, 2019 NCUWM Presentation 18/24

Dihedral Groups Introduction Background F ∗ -diagrams α ( G ) and α P ( G ) Let’s look at an example where α ( D 2 n ) � = α P ( D 2 n ). α P ( G ) Conjectures Assume n = 136 , 191 , 250. In this case, and Theorems α P c ( Z m ) α ( D 2 n ) = 1 , 044 , Dihedral Groups Future Work and α P ( D 2 n ) = 68 , 095 , 625 + 2 . January 27, 2019 NCUWM Presentation 18/24