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Open Problems in Symmetry

Discover semiregular symmetries in graphs, edge and vertex transitive structures, and trivalent and tetravalent graph patterns. Dive into toroidal graphs and inverse holes, with intriguing open problems.

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Open Problems in Symmetry

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  1. Open Problems in Symmetry SIGMAC ‘98 SIGMAC ‘02 SIGMAC ‘06 SIGMAP ‘10 SIGMAP ‘14 SIGMAP ‘18 West Malvern Oaxaca Aveiro Flagstaff Morelia

  2. I: Semiregular symmetries And their diagrams

  3. If G is a graph, a semiregular symmetry of G is a symmetry which acts as one or more cycles of the same length n > 1. For example, in the cube, (1 2 3 4)(5 6 7 8) is a semiregular symmetry But (1 2 3 5 6 7)(4 8) is not.

  4. Diagrams of semiregular symmetries u u u u u u ( ) u 1 2 3 4 5 6 v v v v v v v ( ) 1 2 3 4 5 6 w ( ) w w w w w w 1 2 3 4 5 6 Consider this SRS of order 6: Mod 6 2 2 1

  5. Some questions: Given a diagram, what values of the parameters give an edge-transitive graph? (2) Given k and d, which diagrams on k nodes of degree d allow parameters which give an edge-transitive graph? (3) Ditto the above, for a vertex-transitive graph.

  6. A graph is circulant, bicirculant, tricirculant provided that it has a SRS with exactly 1, 2, 3 cycles.

  7. Circulant trivalent graphs N=4, a=1, tetrahedron Mod N N=6, a=1, K3,3

  8. Bicirculant trivalent graphs Mod N

  9. Bicirculant trivalent graphs (1) Mod N Generalized Petersen Graphs (Frucht, Graver, Watkins) N= 4, a=1, b = 1: cube Q3 N= 5, a=1, b = 2: Petersen N= 8, a=1, b = 3 : Möbius-Kantor N= 10, a=1, b = 2 : Dodecahedron N= 10, a=1, b = 3 : Desargues = B(Petersen) N= 12, a=1, b = 5 : Nauru N= 24, a=1, b = 5 : F48

  10. Bicirculant trivalent graphs (2) Mod N N=2, a = 1, K4

  11. Bicirculant trivalent graphs (3) Mod N N=any, a = 1, b =r, {6,3}B, C for (B, C) = 1. Homework 1: a. Given N and r, find B, C. b. Given (B, C) = 1, find N, r.

  12. Tricirculant trivalent graphs Mod N N = 6, a = 1, b = 2: Pappus N = 18, a = 1, b = 2:{6, 3}3,3 N=10, a = 1, b = 3: 8-cage N = 2, b = 1: K3,3 None Marusic, Kutnar, Kovacs

  13. Circulant tetravalent graphs N=any, a=1 , b2 =±1 mod , b ≠ ±1 mod N Mod N N=2m, a=1, b = m±1

  14. Bicirculant tetravalent graphs Mod N None Rose window graphs, four families, all with a = 1 None Kovacs, Kuzman, Malnic, Wilson

  15. Bicirculant tetravalent graphs Three individual cases N = 7, [a,b,c] = [1,2,4] N = 13 , [a,b,c] = [1,3,9] N = 14 , [a,b,c] = [1,4,6] Mod N Three families: (1) N = any, [a,b,c] = [1,k+1,k2+k+1] for (k+1)(k2+1) = 0 mod N. (2) N = any, [a,b,c] = [1,k, 1-k] for (k-1)(2k) = 0 mod N. (3) N = product of at least 3 different primes, and none of [a,b,c] relatively prime to N.

  16. Tricirculant tetravalent graphs Mod N

  17. Tricirculant tetravalent graphs Mod N

  18. TC1 Toroidal Spidergraph PS(3, n; r) -r2 3 Plus sporadic examples at n = 4, 8, 8, 21

  19. TC8 Toroidal MSY(3, n; a, b) Marusic & Sparl, JACO 2008

  20. TC6 Propellor graphs Matthew Sterns PrN(a, b, c, d) PrN(1, 2d, 2, d) for N even and d2 = ±1 (mod N) 2-weaving Tip->ABABA . . PrN(1, b, b+4, 2b+3) for 4|N and 8b+16 = 0 (mod N) 4-weaving Tip -> ABCBABCBA . .

  21. TC6 Propellor graphs PrN(a, b, c, d) Pr5(1, 1, 2, 2) Pr10(1, 1, 2, 2) Pr10(1, 4, 3, 2) Pr10(1, 1, 3, 3) Pr10(2, 3, 1, 4)

  22. Open Problems: 1: Finish the classification of edge-transitive (or vertex-transitive) tetravalent tricirculant graphs. 2: Tetracirculant tetravalent . . . 3: . . .pentavalent . . 4: How can we tell by looking at a diagram whether it has a nice parameterized family of edge-transitive covers or just sporadic ones?

  23. II: Toroidal Graphs

  24. Maps of type {4,4} Formed from the tessellation {4,4} By factoring out some group T of translations There are three ways to construct an edge-transitive map:

  25. {4,4}b,c T:<(b,c), (-c, b)> {4,4}3,2

  26. {4,4}<b,c> T:<(b,c), (c, b)> {4,4}<3,1>

  27. {4,4}[b,c] T:<(b,b), (-c, c)> {4,4}[3,2]

  28. Open Problem: If some construction gives you a toroidal map or graph, how in the ______ can you tell which one it is?

  29. Example: Toroidal

  30. The diagram (with a = 1) gives: A0 A2 A3 A4 A1 A5 B0 B2 B3 B4 B1 B5 C0 C2 C3 C4 C1 C5 Ab+4 Ab+3 Ab Ab+5 Ab+1 Ab+2 Which values of a, b give an edge-transitive graph? To which families might it belong?

  31. III: Inverse Holes

  32. In a map, a second-order ‘hole’ is a path (a cycle, actually) which encloses two faces of M on the right at every vertex.

  33. Similarly, a third-order ‘hole’ is a path which encloses three faces of M on the right at every vertex.

  34. A j-th order hole in a map: There are j faces on the left at each vertex.

  35. The ‘hole’ operator, Hj(M), gives the map whose faces are the j-th order holes of M. Example: Great Dodecahedron = H2(Icosahedron)

  36. Consider the map {3,6}3,0: Second-order holes:

  37. Then this is the map H2({3,6}3,0): This turns out to be the map {6,3}1,1. Actually, H2 divides the map into 3 copies of {6,3}1,1.

  38. Open Problem: Given a rotary map M and a number j, Find all rotary maps N such that Hj(N) = M. Given a rotary map M and numbers j and k, Find all rotary maps N such that Hj(N) consists of k copies of M.

  39. Example: Given the tetrahedron {3,3} and numbers j = 2 and k = 2, Find all rotary maps N such that H2(N) consists of 2 copies of the tetrahedron.

  40. Example: B B 6 1 12 5 7 11 A 2 8 3 9 A D C C D 10 4 C B B C 5 * 10 8 1 3 9 D A A D 3 11 7 2 4 B B C C *: B A D B A D B A 1 9 5 7 3 11 1

  41. 12 12 5 4 2 1 3 3 10 7 11 10 8 6 6 * 5 1 4 2 9 9 This is the Petrie of Grek’s map.

  42. IV: Mates

  43. Suppose you have a vertex v in a graph Suppose you have an edge e in a graph Suppose you have a cycle C in a graph Suppose you have a vertex v in a map Suppose you have an edge e in a map Suppose you have a face F in a map Suppose you have a Petrie path P in a map Suppose you have a vertex v in a polytope

  44. Suppose you have a vertex v in a graph and suppose the stabilizer of v fixes exactly one other vertex, v’. Call v and v’ mates. Usually, the permutation switching each vertex with its mate is a symmetry of the graph. But . . . . What if it’s not?

  45. Call such a graph . . . . . . wait for it . . un-sym- mate-rical

  46. First example: F26 The underlying graph of {6,3}1,3

  47. Second example: C4[30,8] The Three-Arc Graph of the Petersen Graph The Medial Graph of {4,5}6

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