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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!
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