Cyclic m-Cycle Systems with m ≤ 32 or m=2q with q a Prime Power

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Cyclic m-Cycle Systems with m 32

or m ¼ 2

q

with q a Prime Power

Shung-Liang Wu,1 _{Hung-Lin Fu}2

1_{National United University, Miaoli, Taiwan}

2_{Department of Applied Mathematics, National Chaio Tung University,}

Hsin Chu, Taiwan, E-mail: [email protected]

Received February 16, 2004; revised February 17, 2005

Published online 10 November 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.20082

Abstract: In this paper, the necessary and sufficient conditions for the existence of cyclic 2q-cycle and m-2q-cycle systems of the complete graph with q a prime power and 3 m 32 are given.# 2005 Wiley Periodicals, Inc. J Combin Designs 14: 66–81, 2006.

Keywords: m-cycle system; cyclic

1. INTRODUCTION

For a graph G, let V(G) and E(G) be respectively the vertex set and edge set of G and let S be a collection of cycles of length m (namely, m-cycles) such that each edge in E(G) belongs to exactly one cycle in S. Then the pair (V(G), S) is called an m-cycle

system of G. An m-cycle system of Kv is also referred to as an m-cycle system of

order v. Here, Kvis the complete graph on v vertices. An obvious necessary condition

for the existence of an m-cycle system of Kvis that m v, v is odd, and m divides the

number of edges in Kv.

Alspach and Gavlas [1] and Sˇajna [14] have completely settled the existence

problem of m-cycle systems of Kv and Kv I, where I is a 1-factor.

Let a pair (V, S) be an m-cycle system of Kv and let be an automorphism group

of the m-cycle system (V, S) (i.e., a group of permutations on v vertices leaving the

collection S of cycles invariant). If there is an automorphism 2 of order v, then

Contract grant sponsor: NSC; Contract grant number: 93-2115-M-239-001 # 2005 Wiley Periodicals, Inc.

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the m-cycle system (V, S) is said to be cyclic. For an m-cycle system of Kv, the vertex

set V can be identified with Zv. So, the automorphism can be represented by

: i! i þ 1ðmod vÞ or : ð0; 1; . . . ; v 1Þ on the vertex set V¼ Zv.

In 1938, Peltesohn [10] proved that there exists a cyclic 3-cycle system for each

admissible value of v6¼ 9. Kotzig [9] and Rosa [11,13] showed that for even m, there

exists a cyclic m-cycle system of order 2kmþ 1. Moreover, Rosa [12] also proved that

there exist cyclic m-cycle systems where m¼ 3, 5, 7. Buratti and Del Fra [5], Bryant

et al. [7], and the present authors [8] independently proved that for any integer m with

m 3, there exists a cyclic m-cycle system of order 2km þ 1. Recently, Buratti and

Del Fra [6] present the result that if m is an odd integer with m6¼ 15 and m 6¼ p

where p is prime and > 1, then there exists a cyclic m-cycle system of order 2kmþ m with exception: (m, k)¼ (3, 1). More recently, Vietri [16] has completely filled in the gap created by Buratti and Del Fra [6]. So we have the following results.

Theorem 1.1 [5,7,8]. For any integer m with m 3, there exists a cyclic m-cycle

system of order 2kmþ 1.

Theorem 1.2 [6,16]. Given an odd integer m 3, there exists a cyclic m-cycle

system of order 2kmþ m for any admissible value of k with the only definite

exceptions of (m, k)¼ (3,1), (15,0), and (p_{, 0) with p a prime and > 1.}

The above theorem gives, in particular, a complete answer to the existence question for cyclic q-cycle systems with q a prime power.

With the joint effort of a number of researches [5–13,16], the existence question for cyclic q-cycle systems has been settled for q a prime power. When q is not a prime power, the problem becomes much more difficult and is far from being solved.

In this paper, we settle the existence questions for cyclic 2q-cycle systems with q a

prime power and for cyclic m-cycle systems with m 32.

2. DEFINITIONS AND PRELIMINARIES

Throughout this paper, we shall assume that the vertex set of Kv is Zv and

use (a b) to denote the difference of the edge {a, b} in Kv. Given an m-cycle C¼

ðc0; c1; . . . ; cm1Þ on Kv, let Cþ i ¼ ðc0þ i; c1þ i; . . . ; cm1þ iÞ (mod v), where

i2 Zv.

The cycle orbit of C is the set of distinct m-cycles in the collectionfC þ iji 2 Zvg.

The length of a cycle orbit is its cardinality, i.e., the minimum positive integer k such

that Cþ k ¼ C. A base cycle of a cycle orbit O` is a cycle C 2 O` that is chosen

arbitrarily. Any cyclic m-cycle system of order v is generated from base cycles. For the convenience of notation, we write a cycle k-orbit for a cycle orbit of length k.

A cycle v-orbit of C on Kv is said to be full and otherwise short; and for

convenience, the cycle C is called full or short, respectively.

A cycle C with vertices in Zvis of type d if its stabilizer under the natural action of

Zv has order d. The type of an m-cycle in Zv is a common divisor of m and v. It is

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The following lemma (see [3,4]) is a crucial tool for constructing a cycle of a prescribed type d( >1) in a cyclic m-cycle system.

Lemma 2.1. Let C¼ ðc0; c1; . . . ; cm1Þ be an m-cycle on Kqt satisfying the

following conditions:

(1) For 0 i 6¼ j r 1; ci6 cj (mod t).

(2) The differences between edges fci; ci1g ð1 i rÞ are all distinct.

(3) cr c0¼ t with coprime with q.

(4) cirþj¼ it þ cj (mod qt), where 0 j r 1 and 1 i q 1.

Then C is a cycle of type q and the setfC þ ij0 i < tg forms a cycle t-orbit of C. Consequently, C can be viewed as a base cycle of the cycle t-orbit.

To simplify, the m-cycle C¼ ðc0; c1; . . . ; cr1; t; tþ c1; . . . ; tþ cr1; . . . ;

ðq 1Þt; ðq 1Þt þ c1; . . . ;ðq 1Þt þ cr1Þ in Lemma 2.1 is denoted by C ¼

½c0; c1; . . . ; cr1t in accordance with [6].

For example, the 10-cycle C¼ ð0; 14; 13; 27; 26; 40; 39; 53; 52; 1Þ ¼ ½0; 14₁₃is of type 5 on K65 and the setfC; C þ 1; . . . ; C þ 12g forms the cycle 13-orbit of C.

Proposition 2.2. If m< v < 2mþ 1 and gcd(m, v) is an odd prime power, then no

cyclic m-cycle system of order v exists.

Proof. Suppose, on the contrary, that there exists such a cyclic m-cycle system of

order v, (V, S), set gcd(m,v)¼ p_{(where p is a prime), and let C be the m-cycle of S}

containing the edge f0; v=pg. The hypothesis v<2m implies that jSj ¼

vðv 1Þ=2m < v so that the orbit of C has length smaller than v. Equivalently, the stabilizer of C is not trivial. On the other hand, the type of C is a divisor of pso that the subgroup of Zvof order p (that is, < v/p > ) is certainly contained in the stabilizer

of C. This means that Cþ iv/p ¼ C for i ¼ 0, 1, . . . , p 1 and hence the edges f0; v=pg; fv=p; 2v=pg; . . . ; fðp 1Þv=p; 0g

belong to C. Moreover, it is immediate to see that these edges form the p-cycle (0, v/p, 2v/p, . . . , (p 1)v/p). This is possible only if m ¼ p but, in this case, m would be

a divisor of v so that we would have v¼ m or v > 2m, a contradiction. &

It is worthwhile to note that a cyclic m-cycle system of order less than 2mþ 1 may

exist. As stated previously, Buratti and Del Fra in [6] proved that if m is odd with

m6¼15 and m6¼p_{, where p is prime and > 1, then there exists a cyclic m-cycle}

system of order m.

Throughout this paper, we shall use @C to denote the multiset of partial differences fðci ci1Þji ¼ 1; 2; . . . ; m=dg of an m-cycle C ¼ ðc0; c1; . . . ; cm1Þ of type d

where cm¼ c0. Given a set D¼ fC1; C2; . . . ; Cpg of m-cycles with vertices in Zv,

the list of partial differences from D is the union of the multisets @C1; . . . ;

@Cp; i:e:; @D¼ [pi¼1@Ci.

As a special case of general results concerning graph decompositions with a sharply vertex transitive automorphism group [2], we have:

Lemma 2.3. A set D of m-cycles with vertices in Zvis a set of base cycles of a cyclic

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For each integer m 3, let Spec(m) be the set of v for which there exists an m-cycle system of order v. By [1] and [14] we have SpecðmÞ ¼ f2mt þ wjt 2

N; w2 WðmÞg where W(m) is the set of odd integers w in the open interval (1, 2m)

such that w(w 1)/2 0 (mod 2m). We have:

Proposition 2.4.

(1) If m is an odd prime power, then W(m)¼ {1, m}.

(2) W(15)¼ {1, 15, 21, 25}.

(3) W(21)¼ {1, 7, 15, 21}.

(4) If m is an odd prime power and m 1 (mod 4), then W(2m) ¼ {1, m}.

(5) If m is an odd prime power and m 3 (mod 4), then W(2m) ¼ {1, 3m}.

(6) Wð2k_{Þ ¼ f1gfor k 2.} (7) W(12)¼ {1, 9}. (8) W(20)¼ {1, 25}. (9) W(24)¼ {1, 33}. (10) W(28)¼ {1, 49}. (11) W(30)¼ {1, 21, 25, 45}.

Proof. Applying the Chinese Remainder Theorem, we have thatjWðmÞj ¼ 2n_where

n is the number of odd prime factors of m. This allows us to check immediately all equalities (1–11).

For instance, it is immediate to check that Wð15Þ f1; 15; 21; 25g. On the other

hand, from the above paragraph, W(15) has size 4 so that (2) follows. &

Throughout we shall assume Ciand Cj to be respectively full and short m-cycles

on Kv, and each set of values of the form fc1;c2; . . . ;cng will be denoted by

fc1; c2; . . . ; cng. In particular, the short m-cycle has the form stated in Lemma 2.1.

Proposition 2.5. Let m 2 (mod 8). Then there exists a cyclic m-cycle system of

order v with v m/2 (mod 2m).

Proof. Set v¼ 2pm þ m=2 for p 1. We claim that Ci (1 i p) are full base

cycles and Cjð1 j kÞ are short base cycles.

For i¼ 1; 2; . . . ; p, let Ci¼ ðci;0; ci;1; . . . ; ci;m1Þ be (8k þ 2)-cycles defined as

ci;2j¼ 2j; for 0 j 2k; 8k 2jþ1; for 2kþ 1 j 4k; and ci;2;jþ1¼ 2kð4p þ 1Þ þ i þ 2bði þ 1Þ=2c; for j¼ 0; ð2k 1 jÞð4p 1Þ þ 4k þ 4i 1; for 1 j 2k 1; ð2k 1Þð4p þ 3Þ þ 4i; for j¼ 2k; ð j 2k 1Þð4p 1Þ þ 4k þ 4i 3; for 2k þ 1 j 4k 1; ð2k 1Þð4p þ 1Þ þ 4i þ 1; for j¼ 4k: 8 > > > > < > > > > : We have Sp_i¼1@Ci¼ f2 þ jð4p þ 1Þ; . . . ; ð j þ 1Þð4p þ 1Þ; 2kð4p þ 1Þ þ 1; . . . ; 2kð4p þ 1Þ þ 2pj0 j 2k 1g.

For j¼ 1; 2; . . . ; k, let Cj ¼ ½0; ð2j 1Þð4p þ 1Þ þ 1_4pþ1 and so @Cj¼

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Since ðSp_i¼1@CiÞ [ ð

Sk

j¼1@CjÞ ¼ Zv f0g, the desired result follows from

Lemma 2.3. &

For clearness, we give an example to demonstrate the construction of full even

cycles. Let C1 and C2 be full cycles in a cyclic 18-cycle system of order 81. The

construction of C1 and C2 is shown in Figure 1. Note that the vertices with label 0

stand for the same one. By easy computation, we have @C1[ @C2¼ f2; . . . ; 9;

11; . . . ; 18; 20; . . . ; 27; 29; . . . ; 40g.

Proposition 2.6. Let m 6 (mod 8). Then there exists a cyclic m-cycle system of

order v with v 3m/2 (mod 2m) and v > 3m/2.

Proof. Similarly, set v¼ 2pm þ 3m/2 (p 1) and m ¼ 8k þ 6 (k 0). The proof is

divided into two cases, depending on whether k¼ 0 or k > 0.

Case 1. k¼ 0.

For i¼ 1, 2, . . . , p, let Ci be 6-cycles defined as

Ci¼ ð0; 4ðp þ 1Þ þ i þ 2 ði þ 1Þ 2 ; 2; 2pþ 3 þ 2i; 1; 2i þ 2Þ; if 1 i p 1 and Cp ¼ ð0; 5p þ 4 þ 2 ðp þ 1Þ 2 ; 2; 4pþ 5; 1; 2p þ 2Þ: We haveSp_i¼1@Ci¼ f3; . . . ; 4p; 4p þ 3; . . . ; 6p þ 4g.

The short 6-cycles are: C0 ¼ ½0; 4p þ 14pþ3 and C1¼ ½0; 4p þ 24pþ3; and it

follows that @C0[ @C1 ¼ f1; 2; 4p þ 1; 4p þ 2g.

Case 2. k> 0.

For i¼ 1; 2; . . . ; p, let Ci¼ ðci;0; ci;1; . . . ; ci;8kþ5Þ be ð8k þ 6Þ-cycles given by

ci;2j¼

2j; for 0 j 2k þ 1;

8k 2j þ 5; for 2kþ 2 j 4k þ 2; and

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ci;2jþ1¼

ð2k 1Þð4p þ 3Þ þ 4i þ 3; for j¼ 0;

ð2k 1 jÞð4p þ 1Þ þ 4k þ 4i; for 1 j 2k 1;

ð2k þ 1Þð4p þ 5Þ þ i; for 1 2 and j ¼ 2k;

ð2k þ 1Þð4p þ 5Þ þ 1 þ i þ 2jði1Þ₂ k; for i > 2 and j¼ 2k;

2ð4kp þ 5k þ p þ 2Þ þ 2i; for j¼ 2k þ 1; ðj 2k 2Þð4p þ 1Þ þ 4k þ 4i þ 2; for 2kþ 2 j 4k; ð2k 1Þð4p þ 3Þ þ 4i þ 5; for j¼ 4k þ 1; 2kð4p þ 3Þ þ 3; for i¼ 1 and j ¼ 4k þ 2; 2kð4p þ 3Þ þ 2i þ 3; for i > 1 and j¼ 4k þ 2: 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : If p¼ 1, then @C ¼ f4 þ 7j; . . . ; 7ð j þ 1Þj j ¼ 0; 1; . . . ; 2k 2g [ f14k 2; . . . ; 14kþ 3; 14k þ 6; 14k þ 7; 14k þ 8; 14k þ 10g; if p> 1, then Sp_i¼1@Ci¼ f4 þ ð4p þ 3Þj; . . . ; ð4p þ 3Þð j þ 1Þj j ¼ 0; 1; . . . ; 2k 2g [ fð2k 1Þð4p þ 3Þþ 5þ 4jðp þ 1Þ; . . . ; 2kð4p þ 3Þ þ 3 þ 4jðp þ 1Þj j ¼ 0; 1g [ fð2k þ 1Þð4p þ 3Þ þ 6; . . . ; pð8k þ 6Þ þ 6k þ 4g.

The short (8kþ 6)-cycles are: for j ¼ 0; 1; . . . ; k 1; Cj ¼ ½0; 4p þ 4 þ j ð8pþ

6Þ_4pþ3;Ckþj¼ ½0; 4p þ 5 þ jð8p þ 6Þ4pþ3;C2kþj¼ ½0; 4p þ 6 þ jð8p þ 6Þ4pþ3; C3k

¼ ½0; 2kð4p þ 3Þ þ 4_4pþ3, and C3kþ1½0; ð2k þ 1Þð4p þ 3Þ þ 54pþ3.

We have S3kþ1_j¼0 @Cj ¼ f1 þ ið8p þ 6Þ; 2 þ ið8p þ 6Þ; 3 þ ið8p þ 6Þ; 4p þ 4þ

ið8p þ 6Þ; 4pþ 5 þ ið8p þ 6Þ; 4p þ 6 þ ið8p þ 6Þji ¼ 0; 1; . . . ; k 1g [ fð2k 1Þ ð4p þ 3Þ þ 4; 2kð4p þ 3Þ þ 4; 2kð4p þ 3Þ þ 5; ð2k þ 1Þð4p þ 3Þ þ 5g.

It can be checked thatðSp_i¼1@CiÞ [ ð

S3kþ1

j¼0 @CjÞ ¼ Zv f0g, as desired. &

By virtue of Propositions 2.2, 2.4-(4), (5), 2.5, 2.6, and Theorem 1.1, we have:

Theorem 2.7. If m is a prime power, then there exists a cyclic 2m-cycle system of

order v with the only definite exception of v¼ 3m when m 3 (mod 4).

In next section, we shall deal with the m-cycle systems for m not greater than 32. Since the constructions are different between odd cycles and even cycles, we classify the m-cycle systems into two cases: odd and even.

3. ODD CASES

We begin with introducing two results that are important for constructing odd cycles. Lemma 3.1[8]. Let a, b, c, and r be positive integers with c¼ a þ b and r > c. Then

there exists a cycle C of length 4sþ 3 with the set of differences {a, b, c, r,

rþ 1, . . . , r þ 4s 1}.

Proof. We claim that the cycle C¼ ðv0; v1; . . . ; v2sþ1; v2sþ10; v2s0; . . . ; v10Þ of length

4sþ 3 exists according to the following two cases.

Case 1. Either a or b is odd, say b.

The vertices of C are defined as:

v0¼ 0; for j ¼ 0; 1; . . . ; s; v2jþ1 ¼ a þ 2j; v2jþ10¼ c þ 2j; for j ¼ 1; 2; . . . ; s; v2j¼ a

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Case 2. Both a and b are even.

If r is even, then the vertices of C are given by

v0¼ 0; v1¼ a; v10¼ r þ 4s 2; for j ¼ 1; 2; . . . ; s; v2j¼ a þ r þ 4s 2j þ 1; v2j0

¼ c þ r þ 4s þ 2j 4; v2jþ1¼ a þ 2j; and v2jþ10¼ c þ 4s 2j:

If r is odd, then

v0 ¼ 0; v1 ¼ a; v10¼ r þ 4s 1; for j ¼ 1; 2; . . . ; s; v2j¼ a þ r þ 4s 2j; v2jþ1

¼ a þ 2j; v2j0¼ c þ r þ 4s þ 2j 3; and v2jþ10¼ c þ 4s 2j:

A routine verification can show in each case that @C¼ fa; b; c; r; r þ 1; . . . ;

rþ 4s 1g. &

As an example, we use the method stated above to construct a 15-cycle with the set

of differences f1; 2; 3; 6; . . . ; 17g and a ¼ 2, b ¼ 1, c ¼ 3, r ¼ 6, and s ¼ 3. See

Figure 2.

Next, we consider cycles of length 4sþ 1. Note that Lemma 3.2-(1) is also known

in [8], but for completeness, we give a short proof here.

Lemma 3.2. Let a, b, c, and r be positive integers with c¼ a þ b 1 and r > c.

(1) There exists a cycle C of length 4sþ 1 with the set of differences {a, b, c, r, rþ 1, . . . , r þ 4s 3}.

(2) There exists a cycle C of length 4sþ 1 with the set of differences {a, b, c, r, rþ 1, r þ 2k þ 3, r þ 2k þ 4, . . . , r þ 2k þ 4s 2} where k 0.

Proof.

(1) Let C¼ ðv0; v1; . . . ; v2s; v2s0; v2s10; . . . ; v10Þ be a cycle of length 4s þ 1 whose

vertices are defined as

v0¼ 0; v1 ¼ a; v10¼ c; v2s¼ c þ 2s 3; v2s0¼ c þ r þ 2s 2 þ ",where" ¼ 0

or 1 according as c¼ a þ b þ 1 or a þ b 1; and for i ¼ 1, 2, . . . , s 1, v2i¼

cþ2i 3; v2i0¼ c þ r þ 4s 1 2i; v2iþ1¼ c r 2i 1; and v2iþ10¼c þ 2i.

(2) Using the same method of construction stated above, we can obtain the desired

result. &

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For the case of odd cycle, we also need the crucial help of Skolem sequences and hooked Skolem sequences.

A Skolem sequence of order p is a collection of ordered pairs {(si, ti) | 1 i p,

ti si¼ i} with

Sp

i¼1fsi; tig ¼ f1; 2; . . . ; 2pg; and a hooked Skolem sequence of

order p is still a collection of ordered pairs {(si, ti) | 1 i p, ti si¼ i} with

Sp

i¼1fsi; tig ¼ f1; 2; . . . ; 2p 1; 2p þ 1g.

Theorem 3.3[15].

(1) A Skolem sequence of order p exists if and only if p 0 or 1 (mod 4).

(2) A hooked Skolem sequence of order p exists if and only if p 2 or 3 (mod 4).

In what follows, we will assume {(si, ti) | 1 i p, ti si¼ i} to be a (hooked)

Skolem sequence of order p.

Proposition 3.4. There exists a cyclic 15-cycle system of order v with v 21 or 25

(mod 30) with v > 25.

Proof. By Proposition 2.2, we see that the value of v must be greater than 25. The

proof is divided into two parts: v 21 or 25 (mod 30).

Part 1. v 21 (mod 30).

Let v¼ 30p þ 21 for p 1. If p 0 or 1 (mod 4), by Theorem 3.3-(1), there exists

a Skolem sequence of order p so thatSp_i¼1fi; siþ p; tiþ pg ¼ f1; 2; . . . ; 3pg; and

if p 2 or 3 (mod 4), by Theorem 3.3-(2), there exists a hooked Skolem sequence of

order p so thatSp_i¼1fi; siþ p; tiþ pg ¼ f1; 2; . . . ; 3p 1; 3p þ 1g. This means that

if distinct consecutive integers, say {d þ 1, d þ 2, . . . , d þ 12r} for some integers d and r, are in the set of differences from a 15-cycle, then we can repeatedly utilize Lemma 3.1 and hence, p full 15-cycles are obtained. It is therefore enough to show that there exist short 15-cycles Cjð1 j sÞ such that Zv

Ss

j¼1@Cj f0g

f1; 2; . . . ; 3pgðor f1; 2; . . . ; 3p 1; 3p þ 1gÞ constitutes the desired situation as stated above. This can be done as follows.

Case 1. p 1 (mod 4).

If p¼ 1, then C1¼ [0, 5, 1, 8, 2]17, C2¼ [0, 11, 3, 13, 4]17, and C¼ (0, 2, 20, 4,

26, 6, 32, 8, 9, 30, 7, 24, 5, 17, 3).

We have that @C1[ @C2¼ f4; . . . ; 11; 13; 15g and @C ¼ f1; 2; 3; 12; 14;

16; . . . ; 25g.

If p > 1, then we split the proof into the following three subcases.

Subcase 1: p 5 (mod 12), say p ¼ 12k þ 5 for k 0.

C1¼ ½0; 120k þ 54; 156k þ 71; 120k þ 55; 300k þ 141_120kþ57and C2 ¼ ½0; 120k þ

55; 300kþ 136; 120k þ 56; 300k þ 139_120kþ57.

@ C1 [ @ C2 ¼ f36k þ 16; 36k þ 17; 120k þ 54; 120k þ 55; 180k þ 80; . . . ;

180kþ 85g.

C1¼ [0, 120k þ 102, 300k þ 243, 120k þ 101, 300k þ 244]120kþ97 and C2¼ [0,

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@C1[ @C2¼ {36k þ 28, 36k þ 29, 120k þ 102, 120k þ 103, 180k þ 140, . . . ,

180kþ 145}.

120kþ 139, 300k þ 339, 120k þ 138, 300k þ 340]120kþ137.

@C1[ @C2 ¼ {36k þ 40, 36k þ 41, 120k þ 138, 120k þ 139, 180k þ 200, . . . ,

180kþ 205}.

120kþ 22, 300k þ 60, 120k þ 25, 156k þ 33]120kþ27.

@C1[ @C2 ¼ {36k þ 6, 36k þ 8, 120k þ 21, 120k þ 22, 180k þ 35, . . . ,

180kþ 40}.

120kþ 70, 300k þ 165, 120k þ 69, 156k þ 87]120kþ67.

@C1[ @C2 ¼ {36k þ 18, 36k þ 20, 120k þ 69, 120k þ 70, 180k þ 95, . . . ,

180kþ 100}.

120kþ 106, 300k þ 265, 120k þ 105, 156k þ 137]120kþ107.

@C1[ @C2 ¼ {36k þ 30, 36k þ 32, 120k þ 105, 120k þ 106, 180k þ 155, . . . ,

180kþ 160}.

120kþ 39, 300k þ 91, 120k þ 38, 300k þ 92]120kþ37.

@C1[ @C2 ¼ {36k þ 9, 36k þ 11, 36k þ 12, 36k þ 13, 120k þ 38, 120k þ 39,

180kþ 52, . . . , 180k þ 55}.

C1¼ ½0; 120k þ74; 156k þ 97; 120k þ76; 156k þ 101_120kþ77and C2 ¼ ½0; 120kþ

75; 300kþ 188; 120k þ 76; 300k þ 191_120kþ77.

@C1[ @C2 ¼ f36k þ 21; 36k þ 23; 36k þ 24; 36k þ 25; 120k þ 74; 120k þ 75;

180kþ 112; . . . ; 180k þ 115g.

C1 ¼ ½0; 120k þ 110; 156k þ 146; 120k þ 113; 300k þ 289120kþ117 and C2 ¼

½0; 120k þ 111; 156k þ 148; 120k þ 113; 300k þ 290_120kþ117.

@C1[ @C2 ¼ f36k þ 33; 36k þ 35; 36k þ 36; 36k þ 37; 120k þ 110; 120k þ

111; 180kþ 172; . . . ; 180k þ 175g.

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Let v¼ 30p þ 25 for p 1. Similarly, unless otherwise stated, we just consider here the construction of short 15-cycles.

For i¼ 1, 2, 3, 4, Ci ¼ ½0; 6p þ i; 21p þ j6pþ5, where j¼ 11, 10, 12, or 16

according as i¼ 1, 2, 3, or 4, and S4_i¼1@Ci¼ {6p þ 1, . . . , 6p þ 4, 15p þ 5, . . . ,

15pþ 12}.

For i¼ 1, 2, 3, Ci¼ ½0; 6p þ 1 þ i; 21p þ j_6pþ5, where j¼ 10, 14, or 11 according

as i¼ 1, 2, or 3, and C4¼ ½0; 6p þ 7; 21p þ 17_6pþ5 and soS4_i¼1@Ci¼ {6p þ 2,

6pþ 3, 6p þ 4, 6p þ 7, 15p þ 5, . . . , 15p þ 12}.

For i¼ 1, 2, . . . , p, let Ci be the full 15-cycles. Let C1¼ (0, t1þ 1, t1þ 6p þ 14,

t1þ 2, t1þ 6p þ 13, t1þ 3, t1þ 6p þ 12, t1þ 4, s1 6p 1, 6p 2, 3p 1,

6p 3, 3p, 6p 4, 1) and then @C1¼ {1, s1þ 1, t1þ 1, 3p þ 1, . . . ,

3pþ 4, 6p þ 5, 6p þ 6, 6p þ 8, . . . , 6p þ 13}. The rest of the full 15-cycles are

constructed by the same method described in Part 1.

The proofs of the cases when p 2 (mod 4) and p 3 (mod 4) are analogous to

that in Case 2, so we omit the details.

By routine computation, it can be verified in each case that the union of differences

of the short and full 15-cycles is equal to Zv {0}, and the proof then follows from

Lemma 2.3. &

Proposition 3.5. There exists a cyclic 21-cycle system of order v with v 7 or 15

(mod 42).

Proof. Let v¼ 42p þ 7 or 42p þ 15 for p 1. If p 0 (mod 2), for i ¼ 1, 3, . . . ,

p 1, let

ðai; bi; ciÞ ¼ ði þ 1; siþ p; tiþ pÞ

and

ðaiþ1; biþ1; ciþ1Þ ¼ ði; siþ1þ p; tiþ1þ pÞ:

Suppose that p 1 (mod 4), say p ¼ 4k þ 1. If k ¼ 1, let

ða1; b1; c1Þ ¼ ð2; 13; 14Þ; ða2; b2; c2Þ ¼ ð1; 6; 8Þ; ða3; b3; c3Þ ¼ ð4; 9; 12Þ;

ða4; b4; c4Þ ¼ ð3; 7; 11Þ; and ða5; b5; c5Þ ¼ ð5; 10; 16Þ; and if k > 1;

then let

ðai; bi; ciÞ ¼ ði þ 1; siþ p; tiþ pÞ for odd i 4k 3;

ðaiþ1; biþ1; ciþ1Þ ¼ ði; siþ1þ p; tiþ1þ pÞ for odd i 4k 3;

ða4k1; b4k1; c4k1Þ ¼ ð4k 1; 4k þ 1; 8k þ 1Þ;

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and

ða4kþ1; b4kþ1; c4kþ1Þ ¼ ð4k þ 2; 6k þ 2; 10k þ 3Þ:

Refer to [15, p. 458].

Assume that p 3 (mod 4), say p ¼ 4k 1. If k ¼ 1, then let

ða1; b1; c1Þ ¼ ð1; 5; 7Þ; ða2; b2; c2Þ ¼ ð2; 8; 9Þ; and ða3; b3; c3Þ ¼ ð3; 4; 6Þ;

and if k > 1, then let

ðai; bi; ciÞ ¼ ði þ 1; siþ p; tiþ pÞ for odd i 4k 3;

ðaiþ1; biþ1; ciþ1Þ ¼ ði; siþ1þ p; tiþ1þ pÞ for odd i 4k 3;

and

ða4k1; b4k1; c4k1Þ ¼ ð4k 1; 8k 1; 12k 3Þ:

Clearly, ci¼ aiþ biþ 1 or aiþ bi 1 for 1 i p. Furthermore, it is easy to

check that Sp_i¼1fai; bi; cig ¼ f1; 2; . . . ; 3pg, if p 0 or 3 (mod 4) and

Sp

i¼1fai; bi; cig ¼ f1; 2; . . . ; 3p 1; 3p þ 1g, if p 1 or 2 (mod 4).

Part 1. v¼ 42p þ 7 for p 1.

The proof is divided into four cases, depending on whether p 1, 3, 2, or 0 (mod 4).

C ¼ ½0; 3p; 24p þ 4_6pþ1 and @C¼ f3p; 18p þ 3; 21p þ 4g. The construction

of full 21-cycles is the following.

Subcase 1: p 1 (mod 12).

C1 ¼ ð0; a1; a1þ b1; a1þ b1 3p 2; a1þ b1þ 2; a1þ b1 3p 4; a1 þ b1þ

4; a1þ b1 3p 6; a1þ b1þ 6; a1þ b1 3p 8; a1þ b1þ 8; c1þ 18p þ 12 þ ";

c1þ 8; c1þ 3p þ 11; c1þ 6; c1þ 3p þ 13; c1þ 4; c1þ 3p þ 15; c1þ 2; c1þ 3p þ

17; c1Þ, where " ¼ 0 or 1 according as c1 ¼ a1þ b1þ 1 or a1þ b1 1, and we have

@C1 ¼ fa1; b1; c1; 3pþ 2; . . . ; 3p þ 17; 18p þ 4; 18p þ 5g. Moreover, by Lemma

3.2-(1), the remaining p 1 full 21-cycles C2; . . . ; Cp follows.

Notice that the construction of the full 21-cycles C2; . . . ; Cp in the remainder of

Part 1 is the same as that stated above, so we just indicate the construction of the full 21-cycle C1.

C1 ¼ ð0; a1; a1þ b1; a1þ b1 3p 2; a1þ b1þ 2; a1þ b1 18p 2; a1þ b1þ

4; a1þ b1 18p 4; a1þ b1þ 6; a1þ b1 18p 6; a1þ b1þ 8; c1þ 18p þ 24 þ

"; c1þ 8; c1þ 3p þ 11; c1þ 6; c1þ 18p þ 11; c1þ 4; c1þ 18p þ 13; c1þ 2; c1þ

18pþ 15; c1Þ, where " ¼ 0 or 1 according as c1 ¼ a1þ b1þ 1 or a1þ b1 1, and

we have @C1 ¼ fa1; b1; c1; 3pþ 2; . . . ; 3p þ 5; 18p þ 4; . . . ; 18p þ 17g.

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11; c1Þ, where " ¼ 0 or 1 according as c1 ¼ a1þ b1þ 1 or a1þ b1 1, and it follows

that @C1 ¼ fa1; b1; c1; 3pþ 2; . . . ; 3p þ 11; 18p þ 4; . . . ; 18p þ 11g.

The proof is similar to that in Case 1 and omitted.

Case 3. p 2 (mod 4). C¼ ½0; 3p; 9p þ 3_6pþ1 and @C ¼ f3p; 3p þ 2; 6p þ 3g. Subcase 1: p 2 (mod 12). C1 ¼ ð0; a1; a1þ b1; a1þ b1 3p 4; a1 þ b1 þ 2; a1þ b1 6p 2; a1þ b1þ 4; a1þ b1 6p 4; a1þ b1þ 6; a1 þ b1 6p 6; a1 þ b1þ 8; c1þ 3p þ 15 þ "; c1þ 8; c1þ 3p þ 11; c1þ 6; c1þ 6p þ 11; c1 þ 4; c1 þ 6p þ 13; c1þ 2; c1þ 6p þ

15; c1Þ, where " ¼ 0 or 1 according as c1¼ a1þ b1þ 1 or a1þ b1 1, and so

@C1 ¼ fa1; b1; c1; 3pþ 3; . . . ; 3p þ 8; 6p þ 4; . . . ; 6p þ 15g.

By directly and repeatedly using Lemma 3.2-(1), we then have p full 21-cycles C1; . . . ; Cp.

4; a1þ b1 3p 8; a1þ b1þ 6; a1þ b1 6p þ 2; a1þ b1þ 8; c1þ 6p þ 16 þ ";

7; c1Þ, where " ¼ 0 or 1 according as c1¼ a1þ b1þ 1 or a1þ b1 1, and

@C1 ¼ fa1; b1; c1; 3pþ 3; . . . ; 3p þ 14; 6p þ 4; . . . ; 6p þ 9g.

The proof can be obtained by a method similar to that in Case 3.

Part 2. v¼ 42p þ 15 for p 1.

Case 1. p 1 (mod 4), say p ¼ 4k þ 1.

C¼ ½0; 3p; 6p þ 2; 9p þ 6; 6p þ 3; 27p þ 11; 6p þ 5_14pþ5and @C ¼ f3p; 3pþ

2; 3pþ 3; 3p þ 4; 8p; 21p þ 6; 21p þ 7g.

By Lemma 3.2-(2), we have k full 21-cycles C1; . . . ; Ck with Sk_i¼1@Ci¼

f3p þ 5; . . . ; 3p þ 2k þ 4; 8p þ 1; . . . ; 12p 4g, and by Lemma 3.2-(1), there exist 3kþ 1 full 21-cycles Ckþ1; . . . ; C4kþ1withS4kþ1i¼kþ1@Ci¼ Zv @CSk_i¼1@Ci f0g.

Case 2. p 2 (mod 4), say p ¼ 4k þ 2.

C¼ ½0; 3p; 6p þ 2; 9p þ 7; 6p þ 4; 9p þ 10; 6p þ 614pþ5 and @C ¼ f3p; 3pþ

2; 3pþ 3; 3p þ 4; 3p þ 5; 3p þ 6; 8p 1g.

Similarly, by Lemma 3.2-(2) and Lemma 3.2-(1), there exist kþ 1 full 21-cycles

C1; . . . ; Ckþ1 with Skþ1i¼1@Ci¼ f3p þ 7; . . . ; 3p þ 2k þ 8; 8p; . . . ; 12p þ 7g and

3kþ 1 full 21-cycles Ckþ1; . . . ; C4kþ1 withS4kþ2i¼kþ2@Ci ¼ Zv @C

Skþ1

i¼1 @Ci

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C ¼ ½0; 3p þ1; 6p þ3; 9p þ 8; 6p þ5; 9p þ11; 6p þ7_14pþ5and @C ¼ f3p þ 1;

. . . ; 3pþ 6; 8p 2g.

The p full 21-cycles can be obtained by the analogous method as mentioned in Case 1, so we omit the details.

C¼ ½0; 3p þ 1; 6p þ 3; 9p þ 6; 6p þ 2; 14p þ 7; 35p þ 13_14p_þ5and @C¼ f3p þ

1; 3pþ 2; 3p þ 3; 3p þ 4; 8p þ 5; 21p þ 6; 21p þ 7g.

It is still similar to Case 1, and omitted. &

4. EVEN CASES

Proposition 4.1. There exists a cyclic12-cycle system of order v with v 9 (mod 24).

Proof. Let v¼ 24p þ 9 for p 1.

C¼ [0, 1, 3, 12p þ 8]8pþ3 and for i¼ 1, . . . , p, Ci¼ (0, 10p 4i þ 7, 2,

10p 4i þ 8, 4, 12p 2i þ 9, 5, 4p 4i þ 10, 3, 4p 4i þ 9, 1, t), where t ¼ 3,

if i¼ 1 and t ¼ 6p 2i þ 7, if i > 1.

Proposition 4.2. There exists a cyclic 20-cycle system of order v with v 25 (mod

40) and v > 25.

Proof. Note that by Proposition 2.2, there does not exist a cyclic 20-cycle system of

order 25. Let v¼ 40p þ 25 for p 1.

C1¼ [0, 2, 1, 8p þ 8]8pþ5, C2¼ [0, 6, 1, 8p þ 9]8pþ5, and C3¼ [0, 12p þ 11, 1,

20pþ 14]8pþ5.

For i ¼ 1; . . . ; p; Ci ¼ ð0; 20p 8i þ 19; 2; 20p 8i þ 20; 4; 20p 8i þ 18; 6;

20p 8i þ 21; 8; 12p 4i þ 18; 9; 8p 8i þ 17; 7; 8p 8i þ 14; 5; 8p 8i þ 16; 3;

8p 8i þ 15; 1; 12p 4i þ 12Þ. &

Proposition 4.3. There exists a cyclic24-cycle system of order v with v 33 (mod 48).

Proof. Similarly, we just consider the case when v¼ 48p þ 33 for p 1.

C1¼ ½0; 4p þ 15; 4p þ 11; 8p þ12; 12p þ14; 12p þ 13; 12p þ 11; 16p þ 1416pþ11 and C2¼ ½0; 8pþ31; 4pþ11; 12p þ 28; 8p þ 11; 16p þ 29; 24p þ 48; 20p þ 3016pþ11. C1 ¼ ð0; 8p þ 25; 2; 8p þ 26; 4; 8p þ 33; 6; 8p þ 34; 8; 8p þ 40; 10; 8p þ 31; 11; 4pþ 25; 9; 4p þ 19; 7; 4p þ 18; 5; 4p þ 11; 3; 4p þ 10; 1; 4p 4Þ and for i ¼ 2; . . . ; p; Ci¼ ð0; 8p 4i þ 24; 2; 8p 4i þ 25; 4;18p 4i þ 34; 6; 18p 4i þ 35; 8; 22p 4iþ 33; 10; 24p 2i þ 30; 11; 22p 4i þ 35; 9; 14p 4i þ 40; 7; 14p i þ 39; 5; 4p 4i þ 10; 3; 4p 4i þ 9; 1; 10p 2i þ 34Þ. &

Proposition 4.4. There exists a cyclic 28-cycle system of order v with v 49 (mod

56) and v > 49.

Proof. As mentioned previously, we see that v > 49. Let v¼ 56p þ 49 for p 1.

If p¼ 1, then C1¼ [0, 2, 1, 18]15, Ci¼ [0, 2i þ 6, 1, i þ 17]15 for 2 i 5,

C6¼ [0, 32, 1, 23]15, and C¼ (0, 48, 2, 49, 4, 47, 6, 50, 8, 38, 10, 39, 12, 62, 13, 36,

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If p > 1; then C1¼ ½0; 2; 1; 8p þ 108pþ7; Ci ¼ ½0; 8p þ 12 þ 2i; 1; 8p þ 9 þ i8pþ7

for 2 i 6; and for i ¼ 1; . . . ; p; Ci ¼ ð0; 24p 16i þ 40; 2; 24p 16i þ 41; 4;

24p 16i þ 39; 6; 24p 16i þ 42; 8; 8p 8i þ 24; 10; 8p 8i þ 25; 12; 28p 4i þ

38; 13; 8p 8i þ 22; 11; 8p 8i þ 21; 9; 24p 16i þ 35; 7; 24p 16i þ 32; 5; 24p

16iþ 34; 3; 24p 16i þ 33; 1; 28p 4i þ 28Þ.

Proposition 4.5. There exists a cyclic 30-cycle system of order v with v 21, 25, or

45 (mod 60). Proof. Part 1. v¼ 60p þ 21 for p 1. C¼ ½0; 4p þ 2; 4p þ 1; 8p þ 4; 8p þ 2; 12p þ 6; 12p þ 3; 16p þ 8; 16p þ 3; 20pþ 1120pþ7. If p¼ 1, then C ¼ (0, 30, 2, 31, 4, 38, 6, 39, 8, 46, 10, 47, 12, 53, 14, 24, 13, 36, 11, 35, 9, 28, 7, 27, 5, 20, 3, 19, 1, 14). If p > 1, then C1¼ (0, 28p þ 2, 2, 28p þ 3, 4, 28p þ 10, 6, 28p þ 11, 8, 28p þ 18, 10, 28pþ 19, 12, t1, 14, 4pþ 23, 13, 16p þ 20, 11, 16p þ 19, 9, 16p þ 12, 7, 16pþ 11, 5, 16p þ 4, 3, 16p þ 3, 1, 4p þ 7), and for i ¼ 2, . . . , p, Ci¼ (0, 28p

12iþ 14, 2, 28p 12i þ 15, 4, 28p 12i þ 22, 6, 28p 12i þ 23, 8, 28p 12i þ 30,

10, 28p 12i þ 31, 12, ti, 14, 4iþ 12, 13, 16p 12i þ 32, 11, 16p 12i þ 31, 9,

16p 12i þ 24, 7, 16p 12i þ 23, 5, 16p 12i þ 16, 3, 16p 12i þ 15, 1, 4i þ 1),

where for j¼ 1; . . . ;jðpþ1Þ₂ k; t2j1¼ 30p þ 26 44j þ "1, where "1 ¼ 0 or 1

accord-ing as p 0 or 1 (mod 2) and for j ¼ 1; . . . ; p=2b c; t2j¼ t2j1 "2; where "2¼ 1 or

3 according as p 0 or 1 (mod 2). Part 2. v¼ 60p þ 25 for p 1. If p¼ 1, then C1¼ [0, 14, 11, 16, 15, 19]17, C2¼ [0, 32, 11, 34, 15, 37]17, and C¼ (0, 27, 2, 28, 4, 35, 6, 36, 8, 46, 10, 47, 12, 55, 14, 53, 13, 23, 11, 22, 9, 15, 7, 14, 5, 20, 3, 19, 1, 34). If p > 1; then C1 ¼ ½0; 4p þ 10; 4p þ 7; 8p þ 8; 8p þ 7; 12p þ 712pþ5; C2¼ ½0; 8pþ 20, 4p þ 7, 12p þ 18, 8p þ 7, 16p þ 17]12pþ5. C1¼ (0, 8p þ 15, 2, 8p þ 16, 4, 8p þ 23, 6, 8p þ 24, 8, 26p þ 20, 10, 26p þ 21, 12, t1, 14, 28pþ 25, 13, 26p þ 18, 11, 26p þ 17, 9, 4p þ 15, 7, 4p þ 14, 5, 4p þ 7, 3, 4p þ 6, 1, 26pþ 4) and for i ¼ 2, . . . , p, Ci¼ (0, 8p 4i þ 17, 2, 8p 4i þ 18, 4, 26p 8iþ 18, 6, 26p 8i þ 19, 8, 26p 8i þ 26, 10, 26p 8i þ 27, 12, ti, 14, 28p 2iþ 27, 13, 18p 8i þ 36, 11, 18p 8i þ 30, 9, 18p 8i þ 33, 7, 18p 8i þ 27, 5, 4p 4i þ 9; 3; 4p 4i þ 8; 1; 10p 2i þ 22Þ; where for j ¼ 1; . . . ; bðpþ1Þ₂ c; t2j1¼

30p 4j þ 28 þ "1; where "1 ¼ 0 or 1 according as p 0 or 1 (mod 2) and for

j¼ 1, . . . ; p=2b c, t2j¼ t2j1 "2; where "2¼ 1 or 3 according as p 0 or 1 (mod 2).

Part 3. v¼ 60p þ 45 for p 0.

Note that by Proposition 2.6, we know that there exists a cyclic 30-cycle system of order v with v¼ 60p þ 45 for p 1. It is therefore enough to prove that there exists a cyclic 30-cycle system of order 45. This can be easily given as follows:

C1¼ ½0; 83; C2 ¼ ½0; 163; C3 ¼ ½0; 203; C4¼ ½0; 1; 5; 8; 6; 219; and

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We now have the main result, which is obtained by combining the known results [5–13,16] and the propositions proved in Sections 2, 3, and 4.

Theorem 4.6. If 3 m 32, then there exists a cyclic m-cycle system of order v for

all possible values of v with exceptions of (m, v)¼ (3, 9), (6, 9), (9, 9), (14, 21), (15, 15), (15, 21), (15, 25), (20, 25), (22, 33), (24, 33), (25, 25), (27, 27), and (28, 49).

5. CONCLUDING REMARK

Reviewing the construction of above-mentioned cyclic m-cycle systems, it is clear that the construction of the case when m is even is much easier than that of m odd. We expect that the cyclic m-cycle systems with m even can be solved in the near future. Furthermore, in view of Proposition 2.2, we believe that for any admissible value of v such that m < v < 2mþ 1 and gcd(m, v) is not a prime power, then there exists a cyclic m-cycle system of order v.

ACKNOWLEDGMENT

The authors are grateful to the referees for their valuable comments and suggestions. In particular, we highly appreciate one of the referees who has spent a tremendous effort in referring this paper with great patience and also provides an elegant proof of Proposition 2.2 which improves this paper significantly.

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