irreducible 2-fold cycle systems

Irreducible 2-Fold Cycle Systems

Peter JenkinsCentre for Discrete Mathematics and Computing, Department of Mathematics,The University of Queensland, Queensland 4072, Australia,E-mail: [email protected]

Received March 26, 2004; revised February 23, 2005

Published online 24 June 2005 in Wiley InterScience (www.interscience.wiley.com).

DOI 10.1002/jcd.20068

Abstract: In this paper, it is shown that for any pair of integers (m; n) with 4 � m � n, if there

exists an m-cycle system of order n, then there exists an irreducible 2-fold m-cycle system of

order n, except when (m; n) ¼ (5; 5). A similar result has already been established for the case

of 3-cycles. # 2005 Wiley Periodicals, Inc. J Combin Designs 14: 324–332, 2006

Keywords: irreducible; cycle system; BIBD

1. INTRODUCTION AND PRELIMINARIES

An m-cycle is a simple graph with m vertices, say x0; x1; . . . ; xm�1, and edgeset fxi�1xi j 1 � i � m� 1g [ fx0xm�1g. Such an m-cycle is represented byðx0; x1; . . . ; xm�1Þ or ðxm�1; xm�2; . . . ; x0Þ or any cyclic shift of these. If c is an m-cycle, and x and y are distinct vertices of c such that xy is not an edge of c, then xywill be referred to as a chord of c.

Let �Kn denote the multigraph on n vertices in which each pair of vertices is joinedby exactly � edges. An m-cycle system of order n and index � is a pair ðV ;CÞ whereV is the vertex set of �Kn, and C is a collection of m-cycles whose edge sets partitionthe edges of �Kn. Notice that C may contain duplicate m-cycles if � > 1. An m-cyclesystem of index � is also known as a �-fold m-cycle system. For a given m-cyclesystem, it will be assumed that � ¼ 1 when the index is not explicitly stated.

It is clear that if ðV ;C1Þ and ðV ;C2Þ are m-cycle systems of order n and indices �1

and �2, respectively, then ðV;C1 [ C2Þ is an m-cycle system of order n and index�1 þ �2. Conversely, an m-cycle system ðV;CÞ of order n and index � > 1 is said tobe reducible or decomposable if it is possible to partition C into two subsets, C1 andC2, such that ðV;C1Þ and ðV ;C2Þ are m-cycle systems of order n and indices �1

and �2, respectively, for some pair of positive integers, �1 and �2, with �1 þ �2 ¼ �.

# 2005 Wiley Periodicals, Inc.

324

If no such decomposition is possible, the system is said to be irreducible orindecomposable.

Example 1.1. Let V ¼ Z9 and let C contain the following 6-cycles:

ð0; 4; 3; 2; 1; 5Þ; ð3; 4; 6; 8; 5; 7Þ; ð0; 4; 6; 8; 2; 7Þ;ð0; 2; 4; 1; 3; 6Þ; ð0; 3; 5; 1; 7; 8Þ; ð1; 6; 5; 7; 3; 8Þ;ð2; 5; 8; 4; 7; 6Þ; ð0; 5; 4; 2; 3; 1Þ; ð3; 5; 4; 1; 0; 6Þ;ð3; 0; 2; 1; 7; 8Þ; ð1; 6; 2; 7; 0; 8Þ; ð5; 2; 8; 4; 7; 6Þ:

Then ðV ;CÞ is an irreducible 2-fold 6-cycle system of order 9. The irreducibility ofthe system can be seen by considering the top row of cycles. Each pair of cycles fromthis row has an edge in common; thus, no two of these three cycles could occurtogether in a 1-fold system.

The concept of irreducibility has received considerable attention in the context ofbalanced incomplete block designs or BIBDs. In graph theoretic terms, for positiveintegers k and n with k < n, an ðn; k; �Þ-BIBD is a pair ðV ;BÞ where V is the vertexset of �Kn, and B is a collection of copies of Kk (blocks) whose edge sets partition theedges of �Kn. Much work has been done on the construction of irreducible BIBDswith various parameters. For example, it has been shown [3; 5] that irreducibleðn; k; 2Þ-BIBDs exist for all admissible values of k and n with k � 3 except for thecase ðn; kÞ ¼ ð7; 3Þ. For a survey of related results and construction methods, see [8].

The aim of this paper is to determine for any given cycle length m, the values of nfor which there exists an irreducible 2-fold m-cycle system of order n. Since theaforementioned result for BIBDs solves this problem when m ¼ 3, it will be assumedhenceforth that m � 4. Also, if there does not exist a (1-fold) m-cycle system of aparticular order, then there cannot possibly exist a reducible 2-fold m-cycle system ofthat order. Thus, it is only necessary to consider the values of n for which there existsan m-cycle system of order n.

For a given integer m � 4, let SðmÞ denote the set of positive integers n for whichthere exists an m-cycle system of order n; that is, SðmÞ is the spectrum of m-cyclesystems. It is now known [1; 7] that SðmÞ contains precisely those integers n thatsatisfy the following conditions:

1. n � m if n > 1,2. n is odd, and3. nðn� 1Þ=2m is an integer.

In what follows, it will be shown that for all pairs of integers ðm; nÞ with m � 4and n 2 SðmÞnf1g, there exists an irreducible 2-fold m-cycle system of order n,except when ðm; nÞ ¼ ð5; 5Þ. The method used to construct these irreducible m-cyclesystems is based on the same underlying idea used by Billington in [3] to constructirreducible BIBDs. That is, we shall form a 2-fold m-cycle system of order n bytaking the union of two 1-fold m-cycle systems of order n, and then change severalcycles in such a way as to maintain a 2-fold m-cycle system of order n, but destroy thepossible partition of it into two 1-fold systems.

The notion of a trade is clearly relevant here: Two disjoint collections of m-cycles,T1 and T2, are said to form a Cm-trade, denoted by T1 � T2 or T2 � T1, provided everyedge that occurs in a cycle of T1 or T2 occurs the same number of times in T1 as in T2.

IRREDUCIBLE 2-FOLD CYCLE SYSTEMS 325

Example 1.2. Let u¼ða; b; c; dÞ; v¼ðb; e; d; f Þ; u0 ¼ða; b; e; dÞ, and v0 ¼ ðb; c; d; f Þ.Then fu; vg � fu0; v0g is a C4-trade.

It is clear that if ðV ;CÞ is a �-fold m-cycle system of order n and T1 � T2 is aCm-trade with T1 � C, then ðV ; ðC [ T2ÞnT1Þ is also a �-fold m-cycle system oforder n.

In Section 2, we shall construct irreducible 2-fold m-cycle systems of appropriateorders for some small values of m, and show the nonexistence of an irreducible 2-fold5-cycle system of order 5. In Section 3, a general construction of irreducible 2-fold m-cycle systems, which is applicable for all m � 7 is presented.

2. SMALL VALUES OF m

In this section, it will be shown that for each m 2 f4; 5; 6g and n 2 SðmÞnf1g, thereexists an irreducible 2-fold m-cycle system of order n, except when ðm; nÞ ¼ ð5; 5Þ.The nonexistence of an irreducible 2-fold 5-cycle system of order 5 will be shownfirst.

Lemma 2.1. There is no irreducible 2-fold 5-cycle system of order 5.

Proof. Such a system would contain precisely four 5-cycles, so let C ¼fc1; c2; c3; c4g and suppose for a contradiction that ðV;CÞ is an irreducible 2-fold5-cycle system of order 5. Now, any two cycles in C must have at least one edge incommon. For if two cycles were edge-disjoint, they would necessarily form a 1-fold5-cycle system of order 5; the remaining pair of cycles in C would also form such asystem. Furthermore, it is easy to check that a pair of 5-cycles on the same set of 5vertices cannot have exactly one edge in common. Thus, any two cycles in C musthave at least two edges in common. Suppose that

(i) c1 and c2 both contain the edges e1 and e2;(ii) c1 and c3 both contain the edges e3 and e4; and

(iii) c1 and c4 both contain the edges e5 and e6.

Since every edge occurs in precisely 2 cycles, e1; e2; . . . ; e6 are distinct edges, allof which occur in c1. This is a contradiction because c1 contains only 5 edges. &

The following theorem will be used to create irreducible 2-fold m-cycle systems oforder n for some small values of m and n.

Theorem 2.2. If there exists an m-cycle system ðV;CÞ of order n, such that4 � m � n, and C contains a pair of cycles with exactly one vertex in common, thenthere exists an irreducible 2-fold m-cycle system of order n.

Proof. Let ðV;CÞ be an m-cycle system of order n with the stated property. Notethat the existence of two cycles with exactly one vertex in common implies thatn � 2m� 1. Furthermore, n 6¼ 2m� 1 since

ð2m� 1Þð2m� 2Þ2m

¼ 2m� 3 þ 1

m;

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which is not an integer. Thus, n � 2m. Let a; b; c; d; e; f ; g; x1; x2; . . . ; xm�4; y1;y2; . . . ; ym�3 be 2m distinct elements of V such that C contains the cycles

c1 ¼ ðf ; e; a; d; x1; x2; . . . ; xm�4Þ and

c2 ¼ ða; b; c; y1; y2; . . . ; ym�3Þ:Let ðV; �ðCÞÞ denote the m-cycle system obtained from ðV ;CÞ by applying the

permutation � ¼ ðe b d gÞ to each cycle in C. The collection �ðCÞ then contains thecycles

�ðc1Þ ¼ ðf ; b; a; g; x1; x2; . . . ; xm�4Þ and

�ðc2Þ ¼ ða; d; c; y1; y2; . . . ; ym�3Þ:

Since ðV ;CÞ and ðV; �ðCÞÞ are both m-cycle systems of order n, ðV ;C [ �ðCÞÞ isa 2-fold m-cycle system of order n. In order to modify this system to obtain one whichis irreducible, we replace the cycles c1 and �ðc1Þ with the following two cycles:

u ¼ ðf ; e; a; g; x1; x2; . . . ; xm�4Þ and

v ¼ ðf ; b; a; d; x1; x2; . . . ; xm�4Þ:

It is easy to verify that fu; vg � fc1; �ðc1Þg is a Cm-trade. Thus, if D ¼ðC [ �ðCÞ [ fu; vgÞnfc1; �ðc1Þg, then ðV ;DÞ is a 2-fold m-cycle system of order n.Furthermore, ðV;DÞ is irreducible. For suppose that the cycles in D can be partitionedinto two subsets, D1 and D2, such that ðV;D1Þ and ðV ;D2Þ are both 1-fold m-cyclesystems of order n. The cycles c2 and �ðc2Þ both contain the edge aym�3, so (withoutloss of generality) we must have c2 2 D1 and �ðc2Þ 2 D2. But the cycle v containsthe edge ab, which occurs in c2, and the edge ad, which occurs in �ðc2Þ. Thus,v =2 D1 and v =2 D2, which is a contradiction. &

Corollary 2.3. There exists:

1. An irreducible 2-fold 4-cycle system of order 9,2. An irreducible 2-fold 5-cycle system of order 11,3. An irreducible 2-fold 5-cycle system of order 15, and4. An irreducible 2-fold 6-cycle system of order 13.

Proof. It is sufficient to show that for each case, there exists an m-cycle system ofthe appropriate order containing a pair of cycles with exactly one vertex in common.

1. ðZ9; fði; iþ 1; iþ 5; iþ 3Þ j i 2 Z9gÞ is a 4-cycle system of order 9 , whichcontains the cycles ð0; 1; 5; 3Þ and ð1; 2; 6; 4Þ.

2. ðZ11; fði; iþ 5; iþ 9; iþ 1; iþ 2Þ j i 2 Z11gÞ is a 5-cycle system of order 11 ,which contains the cycles ð0; 5; 9; 1; 2Þ and ð5; 10; 3; 6; 7Þ.

3. ðZ15;CÞ where C contains the cycles

ð0; 1; 2; 3; 4Þ; ð0; 2; 4; 1; 3Þ; ð0; 5; 1; 6; 7Þ;ð0; 6; 2; 5; 8Þ; ð0; 9; 1; 7; 10Þ; ð0; 11; 1; 8; 12Þ;ð0; 13; 1; 10; 14Þ; ð1; 12; 2; 7; 14Þ; ð2; 8; 3; 5; 9Þ;ð2; 10; 3; 6; 11Þ; ð2; 13; 3; 9; 14Þ; ð3; 7; 4; 5; 11Þ;ð3; 12; 4; 6; 14Þ; ð4; 8; 6; 5; 10Þ; ð4; 9; 6; 10; 11Þ;ð4; 13; 5; 12; 14Þ; ð5; 7; 8; 11; 14Þ; ð9; 10; 13; 11; 12Þ;ð8; 10; 12; 13; 14Þ; ð7; 11; 9; 8; 13Þ; ð6; 12; 7; 9; 13Þ;


is a 5-cycle system of order 15, which contains the cycles ð0; 1; 2; 3; 4Þ andð4; 8; 6; 5; 10Þ.

4. ðZ13; fðiþ 3; iþ 4; iþ 2; iþ 5; i; iþ 7Þ j i 2 Z13gÞ is a 6-cycle system of order13 , which contains the cycles ð3; 4; 2; 5; 0; 7Þ and ð9; 10; 8; 11; 6; 0Þ. &

The next two lemmas pertain to the existence of m-cycle systems with holes:For positive integers n and u with n > u, an m-cycle system of order n with a hole ofsize u is a pair ðV;CÞ where V is the vertex set of Kn and C is a collection of m-cycleswhose edge sets partition the edges of Kn � Ku.

Lemma 2.4 ([4]). There exists a 5-cycle system of order n with a hole of size 11 forall n � 21 with n � 1 or 5 ðmod 10Þ.Lemma 2.5 ([6]). If m � 4 is even, and u 2 SðmÞ, then there exists an m-cyclesystem of order n with a hole of size u for all n > u with n � u ðmod 2mÞ.

We are now in a position to present the main result of this section.

Lemma 2.6. For each m 2 f4; 5; 6g and n 2 SðmÞnf1g, there exists an irreducible2-fold m-cycle system of order n, except when ðm; nÞ ¼ ð5; 5Þ.Proof. Lemma 2.1 shows that there is no irreducible 2-fold 5-cycle system oforder 5. Also, from Corollary 2.3 and Example 1.1, we have the following set P ofpairs ðm; nÞ for which there exists an irreducible 2-fold m-cycle system of order n:

P ¼ fð4; 9Þ; ð5; 11Þ; ð5; 15Þ; ð6; 9Þ; ð6; 13Þg:

Now, for each m 2 f4; 5; 6g and n 2 SðmÞnf1g, ðm; nÞ 6¼ ð5; 5Þ, it will be shownthat if ðm; nÞ =2 P then it is possible to construct an m-cycle system ðV ;CÞ of order nwith a hole of size u, where ðm; uÞ 2 P. An irreducible 2-fold m-cycle system of ordern can then be formed by taking two copies of each cycle in C, and filling the hole ofsize u with an irreducible 2-fold m-cycle system of order u. Each value of m will beconsidered in turn.

m ¼ 4 : Note that Sð4Þ ¼ fn 2 Zþ j n � 1 ðmod 8Þg. Let n � 1 ðmod 8Þ, n > 9. ByLemma 2.5, there exists a 4-cycle system ðV;CÞ of order n with a hole ofsize 9.

m ¼ 5 : Note that Sð5Þ ¼ fn 2 Zþ j n � 1 or 5 ðmod 10Þg. Let n > 15 be an integerwith n � 1 or 5 ðmod 10Þ. By Lemma 2.4, there exists a 5-cycle systemðV;CÞ of order n with a hole of size 11.

m ¼ 6 : Note that Sð6Þ ¼ fn 2 Zþ j n � 1 or 9 ðmod 12Þg. Let n > 13 be an integerwith n � 1 or 9 ðmod 12Þ. By Lemma 2.5, there exists a 6-cycle systemðV;CÞ of order n with a hole of size u, where u ¼ 9 if n � 9 ðmod 12Þ, andu ¼ 13 if n � 1 ðmod 12Þ.

This completes the proof. &

3. THE GENERAL CONSTRUCTION

The aim of this section is to show that for any pair of integers m and n with7 � m � n, if there exists an m-cycle system of order n, then it is possible to constructan irreducible 2-fold m-cycle system of order n. Our method of construction will

328 JENKINS

differ slightly according to the structure of the 1-fold system. There are two cases toconsider.

First, we shall consider m-cycle systems, which contain a pair of cycles, c1 and c2,such that c2 contains a chord of c1 and an edge joining one of the vertices of thischord to a vertex not in c1. More formally, an m-cycle system will be said to be ofType (1) if there exists a pair of cycles of the form

c1 ¼ ðx0; x1; . . . ; xd; . . . ; xm�1Þ and c2 ¼ ðz; x0; xd; y1; y2; . . . ; ym�3Þ;

where d 2 f2; 3; . . . ; bm=2cg and z =2 fx1; x2; . . . ; xm�1g. (See Fig. 1.) An m-cyclesystem, which is not of Type (1), will be said to be of Type (2).

Theorem 3.1. If there exists a Type ð1Þ m-cycle system of order n, where7 � m � n, then there exists an irreducible 2-fold m-cycle system of order n.

Proof. Let ðV ;CÞ be a Type (1) m-cycle system of order n. Then C contains a pair ofcycles

c1 ¼ ðx0; x1; . . . ; xd; . . . ; xm�1Þ and

c2 ¼ ðz; x0; xd; y1; y2; . . . ; ym�3Þ;

where d 2 f2; 3; . . . ; bm=2cg and z =2 fx1; x2; . . . ; xm�1g.We define a permutation � on the set V as follows: Let �ðuÞ ¼ u for all

u 2 Vnfx0; x1; . . . ; xm�1g, and

�ðxiÞ ¼

xi i ¼ 0;xm�2 i ¼ 1;xi 2 � i � d � 1;xm�1 i ¼ d;xmþd�2�i d þ 1 � i � m� 3;x1 i ¼ m� 2;xd i ¼ m� 1:

8>>>>>>>><>>>>>>>>:

FIGURE 1. The cycles c1 and c2 (with c2 shown in bold) in a Type (1) m-cycle system.


Let ðV ; �ðCÞÞ denote the m-cycle system obtained from ðV;CÞ by applying thepermutation � to each cycle in C. The collection �ðCÞ contains the following twocycles:

�ðc1Þ ¼ ðx0; xm�2; x2; x3; . . . ; xd�1; xm�1; xm�3; xm�4; . . . ; xdþ1; x1; xdÞ and

�ðc2Þ ¼ ðz; x0; xm�1; �ðy1Þ; �ðy2Þ; . . . ; �ðym�3ÞÞ:

Now ðV;C [ �ðCÞÞ is a 2-fold m-cycle system of order n. In order to modify thissystem to obtain one which is irreducible, we replace the cycles c1 and �ðc1Þ with thefollowing two cycles:

u ¼ ðx0; xd; xd�1; xd�2; . . . ; x1; xdþ1; xdþ2; . . . ; xm�1Þ and

v ¼ ðx0; xm�2; x2; x3; . . . ; xd�1; xm�1; xm�3; xm�4; . . . ; xdþ1; xd; x1Þ:

It is easy to verify that fu; vg � fc1; �ðc1Þg is a Cm-trade. Thus, if D ¼ðC [ �ðCÞ [ fu; vgÞnfc1; �ðc1Þg, then ðV ;DÞ is a 2-fold m-cycle system of order n.Furthermore, ðV ;DÞ is irreducible. For suppose that the cycles in D can be partitionedinto two subsets, D1 and D2, such that ðV ;D1Þ and ðV;D2Þ are both 1-fold m-cyclesystems of order n. The cycles c2 and �ðc2Þ both contain the edge zx0, so (withoutloss of generality) we must have c2 2 D1 and �ðc2Þ 2 D2. But the cycle u containsthe edge x0xd, which occurs in c2, and the edge x0xm�1, which occurs in �ðc2Þ. Thus,v =2 D1 and v =2 D2, which is a contradiction. &

It remains to prove the equivalent result for Type (2) m-cycle systems. Severalprefatory results are needed for this.

Lemma 3.2. If ðV ;CÞ is a Type ð2Þ m-cycle system of order n, and c 2 C, then Ccontains a set of cycles C� such that ðVðcÞ;C�Þ is an m-cycle system of order m.

Proof. Let ðV;CÞ be a Type ð2Þ m-cycle system of order n, and let c ¼ðx0; x1; . . . ; xm�1Þ be a cycle in C. For each i; j 2 f0; 1; . . . ;m� 1g, i 6¼ j, the cycle ofC that contains the edge xixj must have vertex set fx0; x1; . . . ; xm�1g; otherwise ðV ;CÞwould be a Type ð1Þ system. Thus, if C� consists of all cycles of C that contain anedge of the form xixj, then ðVðcÞ;C�Þ is an m-cycle system of order m. &

An m-cycle of the graph Km is known as a Hamilton cycle of Km. Hence, an m-cycle system of order m is equivalent to a decomposition of Km into Hamilton cycles.The necessary and sufficient conditions for the existence of an m-cycle system oforder n given in Section 1 imply that Km has a Hamilton cycle decomposition if andonly if m is odd. Thus, from Lemma 3.2, Type ð2Þ m-cycle systems can exist only forodd values of m. It is also evident that a Type ð2Þ m-cycle system of order n can beviewed as an ðn;m; 1Þ-BIBD, in which each block of size m (that is, each copy of Km)has been decomposed into Hamilton cycles. The following well-known Hamilton cycledecomposition of Km appears in [2], and will be used in the proof of Lemma 3.4.

Theorem 3.3. Let m � 5 be odd. Let V ¼ fx1; x0; x1; . . . ; xm�2g and for eachi 2 f0; 1; . . . ; ðm� 3Þ=2g, let

ci ¼ ðx1; xi; x1þi; xm�2þi; x2þi; xm�3þi; x3þi; . . . ; xmþ12þi; xm�1

2þiÞ;

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where subscripts are reduced modulo m� 1. Then ðV ; fci j 0 � i � ðm� 3Þ=2gÞ is aHamilton cycle decomposition of Km.

Lemma 3.4. There exists an irreducible 2-fold m-cycle system of order m for allodd m � 7.

Proof. First, consider the case when m ¼ 7: Let V1 ¼ Z7 and let C1 contain thefollowing 7-cycles:

ð0; 1; 2; 3; 4; 5; 6Þ; ð0; 1; 3; 5; 6; 4; 2Þ; ð0; 4; 1; 6; 2; 3; 5Þ;ð0; 2; 1; 3; 6; 4; 5Þ; ð0; 3; 4; 1; 5; 2; 6Þ; ð0; 3; 6; 1; 5; 2; 4Þ:

Then ðV1;C1Þ is an irreducible 2-fold 7-cycle system of order 7. The irreducibilityof the system can be seen by noting that each pair of cycles taken from the top rowhas an edge in common; thus, no two of these three cycles could occur together in a1-fold system.

Now let m be an odd integer with m � 9, and let ðV;CÞ be the Hamilton cycledecomposition of Km described in Theorem 3.3. Thus, C contains the cycles

c1 ¼ ðx1; x0; x1; xm�2; x2; xm�3; x3; . . . ; xmþ12; xm�1

2Þ and

c2 ¼ ðx1; x1; x2; x0; x3; xm�2; x4; . . . ; xmþ32; xmþ1

2Þ:

Let ðV; �ðCÞÞ denote the m-cycle system obtained from ðV ;CÞ by applying thepermutation � ¼ ðx2 x4Þ to each cycle in C. The collection �ðCÞ then contains thecycles

�ðc1Þ ¼ ðx1; x0; x1; xm�2; x4; xm�3; x3; xm�4; x2; xm�5; x5; . . . ; xmþ12; xm�1

2Þ

and

�ðc2Þ ¼ ðx1; x1; x4; x0; x3; xm�2; x2; xm�3; x5; xm�4; x6; . . . ; xmþ32; xmþ1

2Þ:

Evidently, ðV ;C [ �ðCÞÞ is a 2-fold m-cycle system of order m. Analogously tothe proof of Theorem 3.1, we transform this system into an irreducible one byreplacing the cycles c2 and �ðc2Þ with the following two cycles:

u ¼ ðx1; x1; x2; xm�2; x3; x0; x4; xm�3; x5; xm�4; x6; . . . ; xmþ32; xmþ1

2Þ

and

v ¼ ðx1; x1; x4; xm�2; x3; x0; x2; xm�3; x5; xm�4; x6; . . . ; xmþ32; xmþ1

2Þ:

It is easy to verify that fu; vg � fc2; �ðc2Þg is a Cm-trade. Thus, if D ¼ðC [ �ðCÞÞ [ fu; vgÞnfc2; �ðc2ÞÞg, then ðV;DÞ is a 2-fold m-cycle system of order n.The irreducibility of this system is evident from the following observations:

1. The cycles c1 and �ðc1Þ both contain the edge x1x0,2. The cycles u and c1 both contain the edge x2xm�2, and3. The cycles u and �ðc1Þ both contain the edge x4xm�3.


Thus, no two of u; c1; and �ðc1Þ could occur together in a 1-fold system. &

The desired result for Type (2) m-cycle systems follows readily from Lemmas 3.2and 3.4.

Theorem 3.5. If there exists a Type ð2Þ m-cycle system of order n, where 7 �m � n , then there exists an irreducible 2-fold m-cycle system of order n.

Proof. Let ðV;CÞ be an m-cycle system of order n, and let c 2 C. By Lemma 3.2, Ccontains a set of cycles C� such that ðVðcÞ;C�Þ is an m-cycle system of order m. LetðVðcÞ;DÞ be an irreducible 2-fold m-cycle system of order m. Finally, let E be thecollection of m-cycles consisting of each cycle in D, and two copies of each cycle inCnC�. It is clear that ðV;EÞ is an irreducible 2-fold m-cycle system of order n . &

4. CONCLUDING REMARKS

Combining the results of Sections 2 and 3, as well as Kramer’s result for the case of3-cycles (see [5]), gives the following theorem:

Theorem 4.1. For any pair of integers ðm; nÞ with 3 � m � n, if there exists an m-cycle system of order n, then there exists an irreducible 2-fold m-cycle system of ordern, except when ðm; nÞ 2 fð3; 7Þ; ð5; 5Þg.

REFERENCES

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332 JENKINS

irreducible 2-fold cycle systems

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