1. 1 Rotational and Cyclic Cycle Systems 聯 合 大 學 吳 順 良

2. 2 Outline: Part 1: Cyclic m-cycle systems 1.1. Introduction 1.2 Known results 1.3. Essential tools 1.4 Constructions 1.5. Extension Part 2: 1-rotational m-cycle systems 2.1. Introduction 2.2 Known results 2.3. Essential tools 2.4 Constructions

3. 3 Part 3: Resolvability 3.1. Introduction 3.2 Known results Part 4: Problems

4. 4 An m-cycle, written (c 0, c 1, , c m-1 ), consists of m distinct vertices c 0, c 1, , c m-1, and m edges {c i, c i+1 }, 0  i  m – 2, and {c 0, c m-1 }. An m-cycle system of a graph G is a pair (V, C) where V is the vertex set of G and C is a collection of m-cycles whose edges partition the edges of G. If G is a complete graph on v vertices, it is known as an m- cycle system of order v. Part 1. Cyclic m-cycle systems 1.1. Introduction

5. 5 The obvious necessary conditions for the existence of an m- cycle system of a graph G are: (1) The value of m is not exceeding the order of G; (2) m divides the number of edges in G; and (3) The degree of each vertex in G is even. For any edge {a, b} in G with V(G) = Z v, By  |a - b| we mean the difference of the edge {a, b}.

6. 6 Example K 9 : V = Z 9 ±1 ±2 ±3 ±4 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (3,7) (4,5) (4,6) (4,7) (4,8) (5,6) (5,7) (5,8) (5,0) (6,7) (6,8) (6,0) (6,1) (7,8) (7,0) (7,1) (7,2) (8,0) (8,1) (8,2) (8,3)

7. 7 Given an m-cycle system (V, C) of a graph G = (V, E) with |V| = v, let  be a permutation on V. For each cycle C = (c 0, , c m-1 ) in C and a permutation  on V, let C  = {(c 0 , , c m-1  )  C  C }. If C  = {C   C  C} = C, then  is said to be an automorphism of (V, C).

8. 8 If there is an automorphism  of order v, then the m-cycle system is called cyclic. For a cyclic m-cycle system, the vertex set V can be identified with Z v. That is, the automorphism  can be represented by  : (0, 1, , v  1) or  : i  i + 1 (mod v) acting on the vertex set V = Z v.

9. 9 An alternative definition: An m-cycle system (V, C) is said to be cyclic if V = Z v and we have C + 1 = (c 0 + 1, , c m-1 + 1) (mod v)  C whenever C  C. The set of distinct differences of edges in K v is Z v \ {0}.

10. 10 Example. K 9 : V = Z 9 ±1 ±2 ±3 ±4 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (3,7) (4,5) (4,6) (4,7) (4,8) (5,6) (5,7) (5,8) (5,0) (6,7) (6,8) (6,0) (6,1) (7,8) (7,0) (7,1) (7,2) (8,0) (8,1) (8,2) (8,3)

11. 11 Example. K 9 : (0, 1, 5, 2) (1, 2, 6, 3) (2, 3, 7, 4) (3, 4, 8, 5) (4, 5, 0, 6) (5, 6, 1, 7) (6, 7, 2, 8) (7, 8, 3, 0) (8, 0, 4, 1)

12. 12 The cycle orbit of C is defined by the set of distinct cycles C + i = (c 0 + i, , c m-1 + i) (mod v) for i  Z v. The length of a cycle orbit is its cardinality, i.e., the minimum positive integer k such that C + k = C. A base cycle of a cycle orbit Ò is a cycle in Ò that is chosen arbitrarily. A cycle orbit with length v is said to be full, otherwise short.

13. 13 Example. K 15 : V = Z 15 ±1 ±2 … ±7 (0, 1, 4) (0, 2, 8) (0, 5, 10) (1, 2, 5) (1, 3, 9) (1, 6, 11) (2, 3, 6) (2, 4,10) (2, 7, 12) (3, 8, 13) (4, 9, 14) (14, 0, 3) (14, 1, 7) m = 3

14. 14 1.2. Known results (1) A cyclic 3-cycle system. (1938, Peltesohn) (2) For even m, there exists a cyclic m-cycle system of order 2km + 1. (1965 and 1966, Kotzig and Rosa) (3) Cyclic m-cycle systems where m = 3, 5, 7. (1966, Rosa) (4) For any integer m with m  3, there exists a cyclic m-cycle system of order 2km + 1. (2003, Buratti and Del Fra, Bryant, Gavlas and Ling, Fu and Wu)

15. 15 (5) A cyclic m-cycle system of order 2km + m, where m is an odd integer with m  15 and m  p  where p is prime and  > 1. (2004, Buratti and Del Fra) (6) A cyclic m-cycle system of order 2km + m, where m is an odd integer with m = 15 and m = p .. (2004, Vietri)

16. 16 Theorem For any integer m with m  3, there exists a cyclic m-cycle system of order 2km + 1. Theorem Given an odd integer m  3, there exists a cyclic m-cycle system of order 2km + m.

17. 17 Note that the above theorems give a complete answer to the existence question for cyclic q-cycle systems with q a prime power. (7) Cyclic m-cycle systems where m = 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30. (Fu and Wu) (8) For cyclic 2q-cycle systems with q a prime power. ( Fu and Wu)

18. 18 1.3. Essential tools Spectrum: a set, Spec(m), of values of v for which the necessary conditions of an m-cycle system are met. Proposition If m = ab with a odd and gcd(a, b) = 1, then v = 2pm + ax 0, where p  0 and x 0 is the least positive integral solution of the linear congruence ax  1 (mod 2b) satisfying ax 0  m.

19. 19 If m has n distinct odd prime factors, then |Spec(m)| = + + … + = 2 n. Example. m = 180 = 2 2  3 2  5 m = 1  180 x 0 = 361 v = 361 m = 3 2  (2 2  5) x 0 = 49 v = 441 m = 5  (3 2  2 2 ) x 0 = 101 v = 505 m = (3 2  5)  (2 2 ) x 0 = 5 v = 225 Spec(180) = {v  v  1, 81, 145, or 225 (mod 360)}

20. 20 Skolem sequences and its generalization.  A Skolem sequence of order n is a collection of ordered pairs {(s i, t i ) | 1  i  n, t i  s i = i} with = {1, 2, , 2n}. Example. {(1, 2), (5, 7), (3, 6), (4, 8)}.

21. 21  A hooked Skolem sequence of order n is a collection of ordered pairs {(s i, t i ) | 1  i  n, t i  s i = i} with = {1, 2, , 2n  1, 2n + 1}. Example. {(1, 2), (3, 5), (4, 7)}

22. 22 Theorem (1) A Skolem sequence of order n exists if and only if n  0 or 1 (mod 4). (2) A hooked Skolem sequence of order n exists if and only if n  2 or 3 (mod 4).

23. 23 How to construct a short m-cycle ?  The number of distinct differences in an m-cycle C is called the weight of C.  Given a positive integer m = pq, an m-cycle C in K v with weight p has index v/q if for each edge {s, t} in C, the edges {s + i  v/q, t + i  v/q } ( mod v) with i  Z q are also in C.

24. 24 Example m = 15 = 5  3 and v = 75 The 15-cycle C = (0, 1, 5, 7, 12, 25, 26, 30, 32, 37, 50, 51, 55, 57, 62) in K 75 with weight 5 (differences  1,  2,  4,  5, and  13) has index 25.

25. 25 Proposition Let m = pq. Then there exists an m-cycle C = (c 0, , c m-1 ) in K v with weight p and index v/q if and only if each of the following conditions is satisfied: (1) For 0  i  j  p  1, c i ≢ c j (mod v/q); (2) The differences of the edges {c i, c i-1 } (1  i  p) are all distinct; (3) c p  c 0 = t  v/q, where gcd (t, q) = 1; and (4) c ip+j = c j + i  t  v/q where 0  j  p  1 and 0  i  q  1.

26. 26 Example. m = 15 = 5  3 and v = 75 The 15-cycle C = (0, 1, 5, 7, 12, 25, 26, 30, 32, 37, 50, 51, 55, 57, 62) = [0, 1, 5, 7, 12] 25 in K 75 with weight 5 (i.e.,  C =  {1, 2, 4, 5, 13}) has index 25, and the set {C, C + 1, , C + 24} forms a cycle orbit of C with length 25 in K 75.

27. 27  Given a set D = {C 1, , C t } of m-cycles, the list of differences from D is defined as the union of the multisets  C 1, ,  C t, i.e.,  D =. Theorem A set D of m-cycles with vertices in Z v is a set of base cycles of a cyclic m-cycle system of K v if and only if  D = Z v \ {0}.

28. 28 Example K 15 : V = Z 15 ±1 ±2 … ±7 m = 3 (0, 1, 4) (0, 2, 8) (0, 5, 10) (1, 2, 5) (1, 3, 9) (1, 6, 11) (2, 3, 6) (2, 4,10) (2, 7, 12) (3, 8, 13) (4, 9, 14) (14, 0, 3) (14, 1, 7)

29. 29 1.4. Constructions ( 一 ) Odd cycles: Lemma Let a, b, c, and r be positive integers with c = a + b and r > c. Then there exists a cycle C of length 4s + 3 with the set of differences  {a, b, c, r, r + 1, , r + 4s - 1}.

30. 30 Example. A 15-cycle with the set of differences  {1, 2, 3, 6, , 17} and a = 2, b = 1, c = 3, r = 6, and s = 3. 2 3 6810121416 1 17151311 9 7

31. 31 Lemma Let a, b, c, and r be positive integers with c = a + b  1 and r > c. (1) There exists a cycle C of length 4s + 1 with the set of differences  {a, b, c, r, r + 1, , r + 4s - 3}. (2) There exists a cycle C of length 4s + 1 with the set of differences  {a, b, c, r, r + 1, r + 2k + 3, r + 2k + 4, , r + 2k + 4s - 2} where k  0.

32. 32 Example. A 13-cycle with the set of differences  {1, 2, 4, 5, , 14} and a = 1, b = 2, c = 4, r = 5, and s = 3. 0 13-45-67 4 18616813 1 2791113 6 4 1412108 5

33. 33 Example. m = 15 and v = 81. C 1 = [0, 21, 61, 25, 64] 27 C 2 = [0, 22, 60, 25, 33] 27  C 1   C 2 =  {6, 8, 21, 22, 35, …., 40} Z 81 - {0} - (  C 1   C 2 ) =  {1, 2, 3, 4, 5, 7, 9, …, 20, 23, …, 34}.

34. 34 ( 二 ) Even cycles: Example. m = 18 and K 81. C 1 = [0, 10] 9 and C 2 = [0, 28] 9  C 1   C 2 =  {1, 10, 19, 28} C3:C3:

35. 35 C4:C4:  C 3   C 4 =  {2, …, 9, 11,…, 18, 20, …, 27, 29, …, 40}  C 1   C 2 =  {1, 10, 19, 28} Z 81 – {0} =  C 1   C 2   C 3   C 4

36. 36 1.5. Extension:  If v is even, then there does not exist a cyclic m-cycle system of K v.  K v - I, where I is a 1-factor.  Example. K 8 - I, where I = {(0, 4), (1, 5), (2, 6), (3, 7)}.  Cyclic 4-cycle system of K v – I.

37. 37 Theorem (2003, Wu) Suppose that m 1, m 2, , m r are positive even (odd) integers with = 2 k for k  2. Then there exist cyclic (m 1, m 2, , m r )-cycle systems of K n if and only if n is odd and the value of divides the number of edges in K n. Theorem (2004, Fu and Wu) Suppose that = n. Then there exists a cyclic (m 1, m 2, , m r )-cycle system of order 2n + 1.

38. 38 Part 2. 1-rotational m-cycle systems 2.1. Introduction K v is the graph on v vertices in which each pair of vertices is joined by exactly edges.

39. 39 Given an m-cycle system of G with |V| = v, if there is an automorphism  of order v – 1 with a single fixed vertex, then the m-cycle system is said to be 1-rotatinal. For a 1- rotational m-cycle system, the vertex set V can be identified with {  }  Z v-1. That is, the automorphism  can be represented by  : (  ) (0, 1, , v  2) or  :   , i  i + 1 (mod v - 1) acting on the vertex set V.

40. 40 An alternative definition: An m-cycle system (V, C) is said to be 1-rotational if V = {  }  Z v-1 and we have C + 1 = (c 0 + 1, , c m-1 + 1) (mod v - 1)  C whenever C  C.

41. 41 Example. K 9 : V = {  }  Z 8 ±1 ±2 ±3 ±4 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (3,7) (4,5) (4,6) (4,7) (5,6) (5,7) (5,0) (6,7) (6,0) (6,1) (7,0) (7,1) (7,2)

42. 42 Example. 2K 9 : V = {  }  Z 8 ±1 ±1 ±2 ±2 ±3 ±3 ±4 (0,1) (0,1) (0,2) (0,2) (0,3) (0,3) (0,4) (1,2) (1,2) (1,3) (1,3) (1,4) (1,4) (1,5) (2,3) (2,3) (2,4) (2,4) (2,5) (2,5) (2,6) (3,4) (3,4) (3,5) (3,5) (3,6) (3,6) (3,7) (4,5) (4,5) (4,6) (4,6) (4,7) (4,7) (0,4) (5,6) (5,6) (5,7) (5,7) (5,0) (5,0) (1,5) (6,7) (6,7) (6,0) (6,0) (6,1) (6,1) (2,6) (7,1) (7,0) (7,1) (7,1) (7,2) (7,2) (3,7)

43. 43 2.2. Known results Theorem [2001, Phelps and Rosa] There exists a 1-rotational 3-cycle system of order v if and only if v  3 or 9 (mod 24).

44. 44 Theorem [2004, Buratti] (1) A 1-rotational m-cycle system of K 2pm+1 exists if and only if m is an odd composite number. (2) A 1-rotational m-cycle system of K 2pm+m exists if and only if m is odd with the only definite exceptions: (m, p) = (3, 4t + 2) and (m, p) = (3, 4t + 3).

45. 45 Theorem [2003, Mishima and Fu] If v  0 (mod 2k), then there exists a 1-rotational k-cycle system of K v. Theorem [Wu and Fu] Let q be a prime power and let k be an integer with k = 0 or 1. Then there exist 1-rotational 2 k q-cycle systems of 2K v if and only if 2 k q divides the number of edges in 2K v.

46. 46 2.3. Essential tools Proposition If m = ab with gcd(a, b) = 1, then v = pm + ax 0, where p  0 and x 0 is the least positive integral solution of the linear congruence ax  1 (mod b) satisfying ax 0  m. Given a positive integer m, what is Spec(m) for 2K v ?

47. 47 Proposition [2003, Buratti] Let d i (1  i  m  2) be distinct positive integers with d 1 < d 2 <  < d m-2. Then there exists an m-cycle containing  with difference set  { , , d 1, d 2, , d m-2 }. Proof. Let C m be a full m-cycle defined as C m = ( , 0, a 1, a 2, , a m-2 ), where a i =.

48. 48 Example. Set m = 10 and 1 < 2 < 4 < 5 < 8 < 10 < 12 < 15. Taking -1, 2, -4, 5, -8, 10, -12, 15, C 10 = ( , 0, -1, 1, -3, 2, -6, 4, -8, 7).

49. 49 A Skolem sequence of order n is an integer sequence (s 1, s 2, , s n ) such that = {1, 2, , 2n}. Example. n = 4. {s 1, s 2, s 3, s 4 } = {1, 5, 3, 4}.

50. 50 A hooked Skolem sequence of order n is an integer sequence (s 1, s 2, , s n ) such that = {1, 2, , 2n  1, 2n + 1}. Example. n = 2. {s 1, s 2 } = {1, 3}.

51. 51 2.4. Constructions Example. m = 7 and 2K 21. {s 1, s 2 } = {1, 3}. C 1 = (0, -1, 1, -5, 6, -7, 8) C 2 = (0, -3, 2, -5, 7, -7, 9)  C 1   C 2 =  {1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9} Z 20 – {0} – (  C 1   C 2 ) =  {1, 2, 3, 4, 10} C 3 = ( , 0, -1, 1, -2, 2, -8)

52. 52 Example. m = 10 and 2K 25 C 1 = [0, 5, 1, 4, 2] 12 C 2 = (0, -1, 1, -2, 2, -4, 3, -5, 4, 5)  C 1   C 2 =  {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10} Z 24 – {0} – (  C 1   C 2 ) =  {6, 7, …., 13}.

53. 53 Example. 2K 8 :  1  1  2  2  3  3  4 (0, 4, 6, 2) (0, 2, 4, 6) (0, 1, 4, 5) (0, 1, 4, 5) (1, 5, 7, 3) (1, 3, 5, 7) (2, 3, 6, 7) (2, 3, 6, 7) (2, 6, 0, 4) (1, 2, 5, 6) (1, 2, 5, 6) (3, 7, 1, 5) (3, 4, 7, 0) (3, 4, 7, 0)

54. 54 Part 3. Resolvability 3.1. Introduction A parallel class of an m-cycle system (V, C) of a graph G is a collection of t (= v/m) vertex disjoint m-cycles in C. The m-cycle system is called resolvable if C can be partitioned into parallel classes R 1, , R s such that every vertex of V is contained in exactly one m-cycle of each class.

55. 55 The set R = {R 1, , R s } is called a resolution of the system. A cyclic (1-rotational) m-cycle system is called cyclically (1-rotationally) resolvable if it has a resolution.

56. 56 Example. m = 4 and 2K 28 R 1 : (0,12,25,11) (1,9,2,10) (3,7,4,8) (5,15,6,16) (17,23,18,24) (19,21,20,22) ( ,13,14,26) R 2 : (1,13,26,12) (2,10,3,11) (4,8,5,9) (6,16,7,17) (18,24,19,25) (20,22,21,23) ( ,14,15,27) R 27 : (26,11,24,10) (0,8,1,9) (2,6,3,7) (4,14,5,15) (16,22,17,23) (18,20,19,21) ( ,12,13,25)

57. 57 3.2. Known results  For m even, 1-rotationally resolvable m-cycle systems of K v. (2003, Mishima and Fu)  A cyclically resolvable 4-cycle system of the complete multipartite graph. (Wu and Fu)  A cyclically resolvable 4-cycle system of K v.

58. 58 Part 4. Problems Problem 1: For all even integers m, there exist 1-rotational m-cycle systems of 2K v. Problem 2: For all odd integers m, there exist 1-rotational m-cycle systems of 2K v.

59. 59 Problem 3: For all even integers m, there exist cyclic m-cycle systems of K v. Problem 4: For all odd integers m, there exist cyclic m-cycle systems of K v.

60. 60 Thanks!