Rotational and Cyclic Cycle Systems

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

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

4. Part 1. Cyclic m-cycle systems 1.1. Introduction • An m-cycle, written (c0, c1, , cm-1), consists of m distinct vertices c0, c1, , cm-1, and m edges {ci, ci+1}, 0 im– 2, and {c0, cm-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.

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) = Zv, By |a - b| we mean the difference of the edge {a, b}.

6. Example K9 : V = Z9 ±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. 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 = (c0, , cm-1) in C and a permutation  on V, let C = {(c0, , cm-1)CC }. If C = {CCC} = C, then  is said to be an automorphism of (V, C).

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 Zv. That is, the automorphism  can be represented by • : (0, 1, , v 1) or : ii + 1 (mod v) • acting on the vertex set V = Zv.

9. An alternative definition: • An m-cycle system (V, C) is said to be cyclic if V = Zv and we have C + 1 = (c0 + 1, , cm-1 + 1) (mod v) • C whenever CC. • The set of distinct differences of edges in Kv is Zv \ {0}.

10. Example. K9 : V = Z9 ±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. Example. K9 : (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. The cycle orbit of C is defined by the set of distinct cycles • C + i = (c0 + i, , cm-1 + i) (mod v) • for iZv. • 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. Example. K15 : V = Z15 ±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)

14. 1.2. Known results • A cyclic 3-cycle system. (1938, Peltesohn) • For even m, there exists a cyclic m-cycle system of order 2km + 1. (1965 and 1966, Kotzig and Rosa) • Cyclic m-cycle systems where m = 3, 5, 7. (1966, Rosa) • 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. (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 andm=p.. (2004, Vietri)

16. Theorem For any integer m withm 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. 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. 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 Ifm = abwith a odd andgcd(a, b) = 1, then v= 2pm +ax0, where p  0 andx0is the least positive integral solution of the linear congruence ax 1 (mod 2b) satisfyingax0m.

19. If m has n distinct odd prime factors, then |Spec(m)| = + + … + = 2n. Example. m = 180 = 22325 m = 1180 x0 = 361 v = 361 m = 32(225) x0 = 49 v = 441 m = 5 (3222) x0 = 101 v = 505 m = (325)(22) x0 = 5 v = 225 Spec(180) = {vv 1, 81, 145, or 225 (mod 360)}

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

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

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

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

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 K75 with weight 5 (differences 1,  2,  4, 5, and 13) has index 25.

25. Proposition Let m = pq. Then there exists an m-cycle C = (c0 , , cm-1) in Kvwith weight p and index v/q if and only if each of the following conditions is satisfied: (1) For 0 ijp 1, ci≢cj (mod v/q); (2) The differences of the edges {ci, ci-1} (1 ip) are all distinct; (3) cpc0 = tv/q, where gcd (t, q) = 1; and (4) cip+j = cj + itv/q where 0 jp 1 and 0 iq 1.

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 K75 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 K75.

27. Given a set D = {C1, , Ct} of m-cycles, the list of differences from D is defined as the union of the multisets C1, , Ct, i.e., D = . Theorem A set D of m-cycles with vertices in Zv is a set of base cycles of a cyclic m-cycle system of Kv if and only if D = Zv \ {0}.

28. Example K15 : V = Z15 ±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. 1.4. Constructions (一) Odd cycles: Lemma Let a, b, c, and r be positive integers with c = a + band 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. 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. 6 8 10 12 14 16 2 1 3 17 15 13 11 7 9

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. 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. 1 3 -4 5 -6 7 2 7 9 11 13 1 0 6 4 14 12 10 8 5 4 18 6 16 8 13

33. Example.m = 15 and v = 81. C1 = [0, 21, 61, 25, 64]27 C2 = [0, 22, 60, 25, 33]27 C1 C2= {6, 8, 21, 22, 35, …., 40} Z81 - {0} - (C1C2)= {1, 2, 3, 4, 5, 7, 9, …, 20, 23, …, 34}.

34. (二) Even cycles: Example.m = 18 and K81. C1 = [0, 10]9 and C2= [0, 28]9 C1C2 = {1, 10, 19, 28} C3:

35. C4: C3C4 = {2, …, 9, 11,…, 18, 20, …, 27, 29, …, 40} C1C2 = {1, 10, 19, 28} Z81 – {0} = C1C2 C3C4

36. 1.5. Extension: • If v is even, then there does not exist a cyclic m-cycle system of Kv. • Kv - I, where I is a 1-factor. • Example. K8 - I, where I = {(0, 4), (1, 5), (2, 6), (3, 7)}. • Cyclic 4-cycle system of Kv – I.

37. Theorem (2003, Wu) Suppose thatm1, m2, , mrare positive even (odd) integers with = 2kfork 2.Then there exist cyclic (m1, m2, , mr)-cycle systemsof Kn if and only ifn is oddand the value of dividesthe number of edges in Kn. Theorem (2004, Fu and Wu) Suppose that = n.Then there exists a cyclic (m1, m2, , mr)-cycle systemof order 2n + 1.

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

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 {}  Zv-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. An alternative definition: • An m-cycle system (V, C) is said to be 1-rotational if V = {}  Zv-1 and we have C + 1 = (c0 + 1, , cm-1 + 1) (mod v - 1) C whenever CC.

41. Example. K9 : V = {} Z8 ±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. Example. 2K9 : V = {} Z8 ±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. 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. Theorem [2004, Buratti] (1) A 1-rotational m-cycle system of K2pm+1 exists if and only if m is an odd composite number. (2) A 1-rotational m-cycle system of K2pm+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. Theorem[2003, Mishima and Fu] If v 0 (mod 2k), then there exists a 1-rotational k-cycle system of Kv. 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 2kq-cycle systems of 2Kvif and only if 2kq divides the number of edges in 2Kv.

46. 2.3. Essential tools Given a positive integer m, what is Spec(m) for 2Kv ? Proposition Ifm = abwith gcd(a, b) = 1, then v = pm + ax0, where p  0 andx0is the least positive integral solution of the linear congruence ax 1 (modb) satisfyingax0m.

47. Proposition[2003, Buratti] Let di(1 im 2) be distinct positive integers withd1 < d2 < < dm-2. Then there exists an m-cycle containingwith difference set{, , d1, d2, , dm-2}. Proof. Let Cm be a full m-cycle defined as Cm = (, 0, a1, a2, , am-2), where ai = .

48. Example. Set m = 10 and 1 < 2 < 4 < 5 < 8 < 10 < 12 < 15. Taking -1, 2, -4, 5, -8, 10, -12, 15, C10 = (, 0, -1, 1, -3, 2, -6, 4, -8, 7).

49. A Skolem sequence of order n is an integer sequence (s1, s2, , sn) such that = {1, 2, , 2n}. Example.n = 4. {s1, s2, s3, s4} = {1, 5, 3, 4}.

50. A hooked Skolem sequence of order n is an integer sequence (s1, s2, , sn) such that = {1, 2, , 2n 1, 2n + 1}. Example.n = 2. {s1, s2} = {1, 3}.