Open Problems in Symmetry

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

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