1

Rotational and Cyclic Cycle Systems

聯 合 大 學

吳 順 良

2

Outline:Part 1: Cyclic m-cycle systems

1.1. Introduction1.2 Known results1.3. Essential tools 1.4 Constructions1.5. 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

An m-cycle, written (c0, c1, , cm-1), consists of m distinct ve

rtices c0, c1, , cm-1, and m edges {ci, ci+1}, 0 i m – 2, an

d {c0, cm-1}.

An m-cycle system of a graph G is a pair (V, C) where V is t

he vertex set of G and C is a collection of m-cycles whose ed

ges partition the edges of G.

If G is a complete graph on v vertices, it is known as an m-cy

cle system of order v.

Part 1. Cyclic m-cycle systems

1.1. Introduction

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 m

ean 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 ver

tex set V can be identified with Zv. That is, the automorphis

m 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 w

e 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 minim

um positive integer k such that C + k = C.

A base cycle of a cycle orbit Ò is a cycle in Ò that is chose

n arbitrarily.

A cycle orbit with length v is said to be full, otherwise shor

t.

13

Example.

K15 : V = Z15 ±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

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 2k

m + 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 s

ystem 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 o

dd 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 o

dd integer with m = 15 and m = p.. (2004, Vietri)

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

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 nece

ssary conditions of an m-cycle system are met.

Proposition

If m = ab with a odd and gcd(a, b) = 1, then

v = 2pm + ax0,

where p 0 and x0 is the least positive integral solution of th

e linear congruence ax 1 (mod 2b) satisfying ax0 m.

19

If m has n distinct odd prime factors, then

|Spec(m)| = + + … + = 2n.

Example. m = 180 = 22325

m = 1180 x0 = 361 v = 361

m = 32(225) x0 = 49 v = 441

m = 5 (3222) x0 = 101 v = 505

m = (325)(22) x0 = 5 v = 225

Spec(180) = {v v 1, 81, 145, or 225 (mod 360)}

n

n

1

n

0

n

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)}.

ni ii ts1 },{

21

A hooked Skolem sequence of order n is a collection of ord

ered pairs {(si, ti) | 1 i n, ti si = i} with

= {1, 2, , 2n 1, 2n + 1}.

Example. {(1, 2), (3, 5), (4, 7)}

ni ii ts1 },{

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

How to construct a short m-cycle ?

The number of distinct differences in an m-cycle C is calle

d the weight of C.

Given a positive integer m = pq, an m-cycle C in Kv with w

eight 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 Zq are also in C.

24

Example

m = 15 = 53 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) i

n Kv with weight p and index v/q if and only if each of the foll

owing 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 dist

inct;

(3) cp c0 = tv/q, where gcd (t, q) = 1; and

(4) cip+j = cj + itv/q where 0 j p 1 and 0 i q 1.

26

Example.

m = 15 = 53 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 diff

erences from D is defined as the union of the multisets

C1, , Ct, i.e., D = .t

1i iC

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 + b and r > c.

Then there exists a cycle C of length 4s + 3 with the set of diffe

rences {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.

2

3

6 8 10 12 14 16

117 15 13 11 9 7

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 differ

ences {a, b, c, r, r + 1, , r + 4s - 3}.

(2) There exists a cycle C of length 4s + 1 with the set of differ

ences {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.

0

1 3 -4 5 -6 7

4 18 6 16 8 13

12 7 9 11 13

64

14 12 10 8 5

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 syste

m 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 that m1, m2, , mr are positive even (odd) integers w

ith = 2k for k 2. Then there exist cyclic (m1, m2, , m

r)-cycle systems of Kn if and only if n is odd and the value of

divides the number of edges in Kn.

ri im1

ri im1

Theorem (2004, Fu and Wu)

Suppose that = n. Then there exists a cyclic (m1, m2, ,

mr)-cycle system of order 2n + 1.

ri im1

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 auto

morphism 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

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 syst

em 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 2Kv if and o

nly if 2kq divides the number of edges in 2Kv.

46

2.3. Essential tools

Proposition

If m = ab with gcd(a, b) = 1, then

v = pm + ax0,

where p 0 and x0 is the least positive integral solution of t

he linear congruence ax 1 (mod b) satisfying ax0 m.

Given a positive integer m, what is Spec(m) for 2Kv ?

47

Proposition [2003, Buratti]

Let di (1 i m 2) be distinct positive integers with

d1 < 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 = .

ij j

j d1 )1(

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, s

2, , sn) such that = {1, 2, , 2n}.

Example. n = 4. {s1, s2, s3, s4} = {1, 5, 3, 4}.

ni ii iss1 },{

50

A hooked Skolem sequence of order n is an integer seque

nce (s1, s2, , sn) such that = {1, 2, , 2n 1,

2n + 1}.

Example. n = 2. {s1, s2} = {1, 3}.

ni ii iss1 },{

51

2.4. Constructions

Example.

m = 7 and 2K21 . {s1, s2} = {1, 3}.

C1 = (0, -1, 1, -5, 6, -7, 8)

C2 = (0, -3, 2, -5, 7, -7, 9)

C1 C2 = {1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9}

Z20 – {0} – (C1 C2 ) = {1, 2, 3, 4, 10}

C3 = (, 0, -1, 1, -2, 2, -8)

52

Example.

m = 10 and 2K25

C1 = [0, 5, 1, 4, 2]12

C2 = (0, -1, 1, -2, 2, -4, 3, -5, 4, 5)

C1 C2 = {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10}

Z24 – {0} – (C1 C2 ) = {6, 7, …., 13}.

53

Example.

2K8: 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

Part 3. Resolvability

3.1. Introduction

A parallel class of an m-cycle system (V, C) of a graph G i

s a collection of t (= v/m) vertex disjoint m-cycles in C.

The m-cycle system is called resolvable if C can be partitio

ned into parallel classes R1, , Rs such that every vertex of

V is contained in exactly one m-cycle of each class.

55

The set R = {R1, , Rs} is called a resolution of the syste

m.

A cyclic (1-rotational) m-cycle system is called cyclically

(1-rotationally) resolvable if it has a resolution.

56

Example.

m = 4 and 2K28

R1: (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)

R2: (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)

R27: (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

3.2. Known results

For m even, 1-rotationally resolvable m-cycle systems of Kv.

(2003, Mishima and Fu)

A cyclically resolvable 4-cycle system of the complete multi

partite graph. (Wu and Fu)

A cyclically resolvable 4-cycle system of Kv.

58

Part 4. Problems

Problem 1:

For all even integers m, there exist 1-rotational m-cycle

systems of 2Kv.

Problem 2:

For all odd integers m, there exist 1-rotational m-cycle

systems of 2Kv.

59

Problem 3:

For all even integers m, there exist cyclic m-cycle sy

stems of Kv.

Problem 4:

For all odd integers m, there exist cyclic m-cycle syst

ems of Kv.

60

Thanks!