Packing lambda-fold complete multipartite graphs with 4-cycles

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Packing k-Fold Complete Multipartite Graphs

with 4-Cycles

Elizabeth J. Billington1;_{, Hung-Lin Fu}2;y_{, and C.A. Rodger}3;z

1_{Centre for Discrete Mathematics and Computing, Department of Mathematics,}

The University of Queensland, Brisbane, Qld 4072, Australia. e-mail: ejb@maths.uq.edu.au

2_{Department of Applied Mathematics, National Chiao-Tung University, Hsin-Chu,}

Taiwan, R.O.C. e-mail: hlfu@math.nctu.edu.tw

3_{Department of Discrete and Statistical Sciences, 235 Allison Lab, Auburn University,}

AL 36849–5307, USA. e-mail: rodgec1@auburn.edu

Abstract. A maximum packing of any k-fold complete multipartite graph (where there are k edges between any two vertices in diﬀerent parts) with edge-disjoint 4-cycles is obtained and the size of each minimum leave is given. Moreover, when k=2, maximum 4-cycle packings are found for all possible leaves.

1. Introduction and Preliminaries

A k-cycle packing of a graph G is a set C of edge-disjoint k-cycles in G. Such a packing is maximum ifjCj jC0_{j for all other k-cycle packings C}0_{of G. The leave L}

of a packing is the set of edges of G that occur in no k-cycle of the packing; we also refer to the subgraph induced by the edges in L as the leave. The leave of a maximum packing is referred to as a minimum leave. A k-cycle system of G is a k-cycle packing of G with leave L¼ ;.

Let Kðv1; v2; . . . ; vnÞ denote the complete multipartite graph with vertex set V

partitioned into n parts Vi of size vi for 1 i n, and edge set consisting of all

edges between all vertices in Vi and Vj, for 1 i < j n, but no edges between

any two vertices in the same part. The complete bipartite graph Kðv1; v2Þ is also

denoted by the more common Kv1;v2. If G denotes a simple graph (with no

loops or multiple edges), then kG denotes the multigraph obtained from G by

* This research was supported by Australian Research Council Grant A69701550

This research was supported by NSC Grant 88-2115-M-009-013

z This research was supported by NSF Grant DMS-9531722 and ONR Grant N00014-97-1-1067

Digital Object Identiﬁer (DOI) 10.1007/s00373-004-0601-0

_{Graphs and}

Combinatorics

ÓSpringer-Verlag 2005

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replicating each edge of G precisely k times. The term ‘‘k-fold’’ graph is also used.

The existence problem for k-cycle systems of complete graphs Kn has been

actively studied over the past 35 years, and this recently resulted in a complete solution of the problem by Alspach, Gavlas, and Sˇajna [1,12] that was partially based on some work of Hoﬀman, Lindner and Rodger [7]. Maximum packings of Kn were also found in [1,12] in all cases where the leave is a 1-factor (this

restricts the possible values of n), and have been found for all values of n when k2 f3; 4; 5; 6g [6,8,9,11,13]. For a survey, see [10].

The existence problem for k-cycle systems of complete multipartite graphs, even when k¼ 3, is proving to be an extremely diﬃcult problem to solve, partly because so many diﬀerent kinds of graphs have to be considered. For example, one excellent paper deals exclusively with the case where k¼ 3 and all parts except one have the same size [5]. However, it turns out that this myriad of complete multipartite graphs can be handled when looking for 4-cycle systems [4]. Furthermore, perhaps surprisingly, the existence problem for maximum 4-cycle packings of complete multipartite graphs was also completely solved [3], producing the following result.

Theorem 1.1. Let G be a complete multipartite graph with g vertices of odd degree and m vertices in the largest part containing vertices of odd degree (if such a part exists). If C is a 4-cycle packing of G with leave L then C is a maximum 4-cycle packing if and only if

(i) maxfg=2; mg jLj maxfg=2 þ 3; m þ 3g, or

(ii) G has an odd number of parts, n, all of odd size, with n 5 or 7 (mod 8), in which casejLj ¼ 6 or 5 respectively.

In this paper we extend this work to the case of any k-fold complete multi-partite graph. That is, we solve the problem of ﬁnding a maximum 4-cycle packing of any k-fold complete multipartite graph, for all k > 1. Moreover, in the case k¼ 2, we exhibit not just a single minimum leave, but all possible minimum leaves.

The graph theoretic notation not deﬁned here can be found in [15]. Sets in this paper are considered to be multisets, and the union of sets requires each element to occur the the number of times equal to the sum of the numbers of times it occurs in the sets themselves. If G and H are two vertex-disjoint graphs, then G_ H is formed from the union of G and H by joining each vertex in G to each vertex in H with exactly one edge. It will cause no con-fusion if we also refer to a k-cycle packing C as an ordered pair ðV ; CÞ, where V is the set of vertices on which the cycles in C are deﬁned.

Section 2 deals with the case k¼ 2, while Section 3 deals with k ¼ 3. Then Section 4 completes all remaining values of k. We may summarise our results as follows (see Theorems 1.1, 2.7, 3.4, and 4.10).

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Main Theorem Let G be a complete multipartite graph. Let gðkÞ be the number of vertices of odd degree inkG, and let mðkÞ be the number of vertices in the largest part ofkG containing vertices of odd degree. There exists a maximum 4-cycle packing of kG with some leave Lk satisfyingjLkj ¼ l if and only if

(i) if G=Kð1;nÞ or Kð1; 1; 1Þ; then Lk¼ EðkGÞ,

(ii) if G=Kð1;1;nÞ and n > 1 then Lk¼ kK2_ K1

if n and k are odd; and kK2 otherwise;

(iii) ifgðkÞ ¼ 0, jEðkGÞj 1 (mod 4), and G 6¼ Kð1; 1; nÞ, then jLkj ¼ 5,

(iv) ifgðkÞ ¼ 0, jEðkGÞj 2 (mod 4), and k ¼ 1, then jLkj ¼ 6, and otherwise

(v) l is the unique integer satisfying

(1) maxfgðkÞ=2; mðkÞg l maxfgðkÞ=2 þ 3; mðkÞ þ 3g, and (2) 4 dividesjEðkGÞj l.

The following result addresses the necessity of the conditions (i)–(v) for the existence of a maximum 4-cycle packing of kG in the Main Theorem.

Given that the lower bound in condition (v)(1) and condition (v)(2) are proved below to be necessary conditions for the existence of a maximum packing, it is clear that any 4-cycle packing which also satisﬁes the upper bound in con-dition (v)(1) must be a maximum 4-cycle packing.

Lemma 1.2. If there exists a maximum 4-cycle packing C of kG then, (referring to conditions in the Main Theorem above), C satisiﬁes conditions (i) and (ii), C has a leave Lkwhich must satisfyjLkj 5 or 6 in conditions (iii) and (iv) respectively, and

C satisﬁes the lower bound in condition (v(1)) and condition (v(2)) of the Main Theorem.

Proof. Let C be a maximum 4-cycle packing of kG, and let L be its leave. Each 4-cycle in C accounts for either 0 or 2 edges incident with each vertex in kG, and therefore each vertex of odd degree in kG must have odd degree in L. Therefore jLj gðkÞ=2, and since clearly L contains no edge joining two ver-tices in the same part of kG, the condition jLj mðkÞ also follows. Therefore jLj maxfgðkÞ=2; mðkÞg. Also, each 4-cycle in C accounts for four edges in kG, so clearly 4 divides jEðkGÞj jLj. So the lower bound in condition ðvð1ÞÞ is necessary, as is condition ðvð2ÞÞ.

If G¼ Kð1; nÞ or G ¼ Kð1; 1; 1Þ ¼ K3then kG contains no 4-cycles, so C¼ ; is

the only 4-cycle packing of G. So condition ðiÞ is necessary.

Similarly, if G¼ Kð1; 1; nÞ then kG contains no 4-cycles which contain an edge joining the vertices in V1[ V2. So since 4 divides jEðkGÞj jLj, it follows that

conditionðiiÞ is necessary.

If all vertices in L have even degree, then clearly jLj 6¼ 1. So if jEðkGÞj 1 (mod 4) thenjLj 5 since jLj jEðkGÞj (mod 4). So condition ðiiiÞ is necessary.

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Similarly, if all vertices in L have even degree and k¼ 1, then jLj 6¼ 2 (since G contains no multiple edges). So if jEðkGÞj 2 (mod 4) then jLj 6 since jLj jEðkGÞj (mod 4). So condition ðivÞ is necessary. (

There is one further result that we need.

Lemma 1.3. There exists a 4-cycle system of kKx;y if and only if minfx; yg 2, kx

and ky are even, and 4 divides kxy.

Proof. This is easy to prove, and also follows from a more general result of

Sotteau [14]. (

We shall use Lemma 1.3 often, so we adopt the following notation. Let BkðX ; Y Þ denote a 4-cycle system of kKx;ywith bipartitionfX ; Y g of the vertex set,

where x¼ jX j and y ¼ jY j. Often the value of k will be clear from the context, in which case we shall simply use BðX ; Y Þ.

Subsequently we also use the existence of a 4-cycle decomposition of kKvfor

appropriate k and v; see [2].

2. The Case k¼ 2

We begin with some useful lemmas.

Lemma 2.1. There exists a 4-cycle packing of 2K(1,1,1,2) with leave ffu1; u2g;

fu1; u2gg in each of the following cases:

(a) u12 V1 and u22 V2, and

(b) u12 V1 and u22 V4.

Proof. Let Vi¼ fig for 1 i 3 and V4 ¼ f4; 5g. Then ðf1; 2; 3; 4; 5g; BÞ is the

required packing, where in case ðaÞ,

B¼ fð1; 3; 2; 4Þ; ð1; 3; 2; 5Þ; ð1; 4; 3; 5Þ; ð2; 4; 3; 5Þg; and in caseðbÞ,

B¼ fð1; 2; 4; 3Þ; ð1; 2; 3; 5Þ; ð1; 3; 2; 5Þ; ð2; 4; 3; 5Þg:

(

Lemma 2.2. There exists a 4-cycle packing of 2K(1,1,2,2) with leave ffu1; u2g;

(a) u12 V1 and u22 V2,

(b) u12 V1 and u22 V3, and

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Proof. Let V1¼ f1g, V2¼ f2g, V3¼ f3; 4g and V4 ¼ f5; 6g. Then ðf1; 2; 3; 4; 5; 6g;

BÞ is the required packing, where in case ðaÞ,

B¼ fð1; 3; 2; 4Þ; ð1; 3; 2; 4Þ; ð1; 5; 2; 6Þ; ð1; 5; 2; 6Þ; ð3; 5; 4; 6Þ; ð3; 5; 4; 6Þg; in caseðbÞ, B¼ fð1; 2; 3; 5Þ; ð1; 2; 5; 4Þ; ð1; 4; 2; 6Þ; ð1; 5; 4; 6Þ; ð2; 3; 6; 4Þ; ð2; 5; 3; 6Þg; and in caseðcÞ, B¼ fð1; 2; 3; 6Þ; ð1; 2; 6; 3Þ; ð1; 3; 2; 4Þ; ð1; 4; 2; 5Þ; ð1; 5; 4; 6Þ; ð2; 5; 4; 6Þg: (

Lemma 2.3. There exists a 4-cycle packing of 2K(3,3) with leave 2K(1,1). Proof. A suitable packing is given byððf1; 2; 3g [ f4; 5; 6gÞ; BÞ where

B¼ fð1; 5; 2; 6Þ; ð1; 5; 3; 6Þ; ð2; 4; 3; 5Þ; ð2; 4; 3; 6Þg:

The leave here isff1; 4g; f1; 4gg. (

Lemma 2.4. There exists a 4-cycle packing of 2K(1,3,2) with leave ffu1; u2g;

(b) u12 V2 and u22 V3.

Proof. Let V1¼ f1g, V2¼ f2; 3; 4g and V3¼ f5; 6g. Then ðf1; 2; 3; 4; 5; 6g; BÞ is

the required packing, where in caseðaÞ,

B¼ fð1; 3; 5; 4Þ; ð1; 3; 6; 4Þ; ð1; 5; 2; 6Þ; ð1; 5; 3; 6Þ; ð2; 5; 4; 6Þg; and in caseðbÞ,

B¼ fð1; 2; 6; 3Þ; ð1; 2; 6; 4Þ; ð1; 3; 5; 4Þ; ð1; 5; 3; 6Þ; ð1; 5; 4; 6Þg:

(

(b) u12 V1 and u22 V3.

Proof. Let V1¼ f1; 2; 3g, V2¼ f4; 5; 6g and V3¼ f7; 8g. Then ðf1; 2; . . . ; 8g; BÞ is

the required packing, where in case ðaÞ, B ¼ B1[ BðV1[ V2; V3Þ, where

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B¼ B2[ Bðf5; 6g; V1[ V3Þ where ððV1;f4g; V3Þ; B2Þ is a 4-cycle packing of

2Kð3; 1; 2Þ (see Lemma 2.4(b)). (

(a) u12 V1 and u2 2 V3, and

(b) u12 V2 and u22 V3.

Proof. Let V1¼ f1g, V2¼ f2; 3; 4g and V3¼ f5; 6; 7g. Then ðf1; 2; . . . ; 7g; BÞ is the

required packing, where in case ðaÞ,

B¼ fð1; 3; 5; 4Þ; ð1; 3; 6; 4Þ; ð1; 5; 2; 6Þ; ð1; 5; 3; 7Þ; ð1; 6; 4; 7Þ; ð2; 5; 4; 7Þ; ð2; 6; 3; 7Þg; and in caseðbÞ,

B¼ fð1; 2; 6; 3Þ; ð1; 2; 7; 4Þ; ð1; 3; 5; 4Þ; ð1; 5; 3; 7Þ; ð1; 5; 4; 6Þ; ð1; 6; 3; 7Þ; ð2; 6; 4; 7Þg: (

Theorem 2.7. Let G¼ 2Kðv1; v2; . . . ; vsþtÞ be the 2-fold complete multipartite graph

with parts V1; V2; . . . ; Vsþt, where vi¼ jVij is odd for 1 i t and is even for

tþ 1 i s þ t. Let V ¼[

sþt

i¼1

Vi and v¼ jV j. There exists a 4-cycle packing of G

with leave L that is a maximum packing if and only if

(a) L=E(G) if G = Kð1;nÞ or G = K(1,1,1), and otherwise (b) if t 0 or 1 (mod 4) then L ¼ ;, and

(c) if t 2 or 3 (mod 4) then L ¼ ffu1; u2g; fu1; u2gg, where

(i) if G = Kð1,1,nÞ then each of u1 and u2 is in a part of size1;

(ii) if G = Kð1, 2n+1; 2Þ then exactly one of u1or u2 occurs in the part of

size2n+1; and

(iii) for all other G, u1 and u2 occur in any two diﬀerent parts.

Proof. We begin by showing that no 4-cycle packings of G with smaller leaves exist, and that ifjLj ¼ 2 then no other choices for u1 and u2 are possible.

If t 2 or 3 (mod 4) then jEðGÞj 2 (mod 4), so necessarily jLj 2. If G¼ Kð1; nÞ or G ¼ Kð1; 1; 1Þ, then G contains no 4-cycles, so clearly L ¼ EðGÞ. If jLj ¼ 2 then since each vertex in G has even degree, L ¼ ffu1; u2g; fu1; u2gg for

some vertices u1 and u2 which, being adjacent vertices, must occur in diﬀerent

parts.

If G¼ Kð1; 1; nÞ and n > 1 with V1¼ fu1g and V2¼ fu2g then no 4-cycle

contains an edge joining u1 to u2. So L¼ ffu1; u2g; fu1; u2gg, since jLj ¼ 2.

If G¼ Kð1; 2n þ 1; 2Þ with say V1¼ fu1g and V3¼ fu2; zg then there is no

4-cycle in Gn ffu1; u2g; fu1; u2gg that contains an edge fu1; zg. But this cannot

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So we now turn to the construction of a maximum 4-cycle packing of G with leave L that satisﬁes conditions ðaÞ; ðbÞ and ðcÞ. Clearly we can assume that sþ t 2, and by ðbÞ we can assume that v 4 and that if s þ t ¼ 2 then minfv1; v2g > 1. We consider various situations in turn.

Case 1. Suppose that t 0 or 1 (mod 4).

For 1 i t, let Wi Vi withjWij ¼ 1; for t þ 1 i s þ t it is convenient to

deﬁne Wi¼ ;. Let W ¼

[t i¼1

Wi, and let ðW ; B1Þ be a 4-cycle system of 2Kt. For

1 i s þ t, clearly we have that jVin Wij is even (possibly zero), and if

Xi¼ ð [ sþt j¼iþ1 VjÞ [ ð [ i1 j¼1

WjÞ, then jXij 2 (by the assumptions on s, t and v).

There-fore, by Lemma 1.3, we can deﬁne the set of 4-cycles B0¼

[

sþt

i¼1

BðVin Wi; XiÞ;

(where we take Bð;; XiÞ ¼ ;). Then ðV ; B1[ B0Þ is the required 4-cycle system of G.

Case 2. Suppose that t 2 or 3 (mod 4).

Let ui2 Vai for 1 i 2; since it only matters if aiis in a part of even or odd

size, we can assume that ða1;a2Þ 2 fð1; 2Þ; ð1; t þ 1Þ; ðt þ 1; t þ 2Þg for notational

convenience; so in particular a1<a2. In each of the following cases, let B0be as

deﬁned in Case 1, where the sets Wi are deﬁned below. Also, as in Case 1, we

assume that Wi¼ ; unless otherwise deﬁned.

(a) Suppose that either t 4, or t ¼ 3 and s 1. For 1 i t let Wi Viwith

jWij ¼ 1, where for 1 j 2 we choose Wj¼ fujg if aj t. Let W ¼St_i¼1Wi. We

consider three cases in turn.

Suppose a2 t and t 4. Let ðW ; B1Þ be a 4-cycle packing of 2Kt with leave

ffu1; u2g; fu1; u2gg. It follows that ðV ; B1[ B0Þ is the required maximum 4-cycle

packing of G.

Suppose a1 t and a2 > t, or a2 t and t ¼ 3. Since t 2 0 or 1 (mod 4), let

ðSti¼3Wi; B1Þ be a 4-cycle system of 2Kt2. Since a2> t or t¼ 3, we know that

s 1, so choose Wtþ1 Vtþ1 with jWtþ1j ¼ 2 and with u22 Wtþ1 if a2 > t. Let

ððW1; W2; W3; Wtþ1Þ; B2Þ be a 4-cycle packing of 2Kð1; 1; 1; 2Þ with leave

ffu1; u2g; fu1; u2gg (see Lemma 2.1 (a) or (b) if u2 2 V2 or u2 2 Vtþ1 respectively).

Then ðV ; B2[ B1[ B0[ BðW1[ W2[ Wtþ1;Sti¼4WiÞÞ is a maximum 4-cycle

pack-ing of G with leaveffu1; u2g; fu1; u2gg.

Finally, suppose that a1> t. LetðSti¼3Wi; B1Þ be a 4-cycle system of Kt2. For

1 j 2 choose Wtþj Vtþj with jWtþjj ¼ 2 and with uj2 Wtþj. Let

ððW1; W2; Wtþ1; Wtþ2Þ; B2Þ be a 4-cycle packing of 2Kð1; 1; 2; 2Þ with leave ffu1; u2g;

fu1; u2gg (see Lemma 2.2(c)). Now ðV ; B2[ B1[ B0[ BðW1[ W2[ Wtþ1[

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(b) Suppose that t¼ 3 and s ¼ 0. If two of the parts have size 1, then by (c)(i), V1[ V2¼ fu1; u2g, so ðV ; BðV1[ V2; V3ÞÞ is the required maximum 4-cycle

packing of G. Otherwise at most one of v1, v2 or v3 is 1. Therefore we can select

three vertices from each of two parts and one vertex from the third part, ensuring that u1 and u2 are among the selected vertices; for 1 i 3 let Wi be the set of

vertices selected from Vi. Let ððW1; W2; W3Þ; B1Þ be a 4-cycle packing of

2KðjW1j; jW2j; jW3jÞ with leave ffu1; u2g; fu1; u2gg (see Lemma 2.6 (a) or (b)). Then

ðV ; B1[ B0Þ is the required 4-cycle packing of G.

(c) Finally, suppose that t¼ 2. If s ¼ 0 then minfv1; v2g 3, so for 1 i 2

let Wi Vi with ui2 Wi and jWij ¼ 3. Let ððW1; W2Þ; B1Þ be a 4-cycle packing of

2Kð3; 3Þ with leave ffu1; u2g; fu1; u2gg (see Lemma 2.3). Then ðV ; B1[ B0Þ is the

required maximum 4-cycle packing.

If s 2 then for 1 i 4 let Wi Vi with jW1j ¼ jW2j ¼ 1, jW3j ¼ jW4j ¼ 2,

andfu1; u2g S4i¼1Wi. By Lemma 2.2(a), (b) or (c) there exists a 4-cycle packing

ððW1; W2; W3; W4Þ; B1Þ of 2Kð1; 1; 2; 2Þ with leave ffu1; u2g; fu1; u2gg. Then

ðV ; B1[ B0Þ is the required maximum 4-cycle packing.

Finally, suppose that s¼ 1. If G ¼ 2Kð1; 1; nÞ then condition (c)(i) requires u1and u2to be in the parts of size 1, soðV ; Bðfu1; u2g; V n fu1; u2gÞÞ is the required

maximum 4-cycle packing.

Otherwise, we now assume that v2 3. If u12 V1and u22 V2, or if u12 V2and

u22 V3, then let W1 V1 withjW1j ¼ 1, W2 V2 withjW2j ¼ 3, and W3 V3 with

jW3j ¼ 2, where fu1; u2g S3i¼1Wi. By Lemma 2.4 (a) or (b), there exists a 4-cycle

packing ððW1; W2; W3Þ; B1Þ of 2Kð1; 3; 2Þ with leave ffu1; u2g; fu1; u2gg, so

ðV ; B1[ B0Þ is the required maximum 4-cycle packing. It remains to consider the

possibility of u12 V1 and u2 2 V3, in which case v1 3 by condition (c)(ii). For

1 i 3 let Wi Vi withjWij ¼ 2 or 3 if i ¼ 3 or i 2 respectively. By Lemma

2.5(b) there exists a 4-cycle packing ððW1; W2; W3Þ; B1Þ of 2Kð3; 3; 2Þ with leave

ffu1; u2g; fu1; u2gg, so ðV ; B1[ B0Þ is the required maximum 4-cycle packing. (

3. The Case k¼ 3

In this section we shall use the following notation. A maximum 4-cycle packing of the complete multipartite graph kG with vertex set V will be denoted byðV ; BkÞ,

with leave Lk, for k¼ 1; 2; 3.

We shall also refer to the graph D where VðDÞ ¼ fa1; a2; u1; u2; u3; u4g and

EðDÞ ¼ ffa1; a2g; fa1; u2g; fa1; u4g; fa2; u1g; fa2; u3gg as Dða1; a2; u1; u2; u3; u4Þ.

This graph is sometimes a subgraph of a leave L1.

We begin with three lemmas.

Lemma 3.1. If there exists a 4-cycle packing ðV ; B1Þ of a complete multipartite

graph G with leave L1such that D L1, and if there exists a 4-cycle packingðV ; B2Þ

of2G with leaveffu1; u2g; fu1; u2gg, then there exists a 4-cycle packing of 3G with

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Proof. Let B3 ¼ B1[ B2[ fða1; a2; u1; u2Þg, so that

L3¼ ðL1n EðDÞÞ [ ffa1; u4g; fa2; u3g; fu1; u2gg and jL3j ¼ jL1j 2:

( Lemma 3.2. If there exists a 4-cycle packing ðV ; B1Þ of a complete multipartite

graph G with leave L1 such that

(i) Q¼ ffu5; u1g; fu5; u2g; fu5; u3g; fu5; u4gg L1 and

(ii) b¼ ðu1; u2; u3; u4Þ 2 B1,

and if there exists a4-cycle packingðV ; B2Þ of 2G with leave ffðu1; u2g; fðu1; u2gg,

then there exists a4-cycle packing of 3G with leave L3 wherejL3j ¼ jL1j 2.

Proof. Let B3¼ ðB1n fbgÞ [ B2[ fðu1; u2; u3; u5Þ; ðu1; u2; u5; u4Þg. Then L3¼

L1n Q [ ffu1; u2g; fu3; u4gg. (

Lemma 3.3. Let Hð1; 2; . . . ; 12Þ be the graph with vertex set f1; 2; . . . ; 12g and edge set EðH Þ ¼ ff6; 10g; f6; 10g; f1; 2g; f1; 3gg [ ffi; jg j 3 i 6; 7 j 12g[ ffi; jg j 7 i 10; 11 j 12g [ ff2; jg j 4 j 12g: There exists a 4-cycle packing BðH ð1; 2; . . . ; 12ÞÞ of Hð1; 2; . . . ; 12Þ with leave the matching L¼ ff3; 11g; f4; 8g; f5; 9g; f6; 10g; f7; 12gg saturating the vertices of odd degree. Proof. The 4-cycles in f(1,2,8,3), (2,4,9,6), (2,5,11,7), (2,9,11,10), (2,11,8,12), (3,7,4,12), (3,9,12,10), (4,10,6,11), (5,7,6,8), (5,10,6,12)g provide the required

4-cycle packing. (

Theorem 3.4. Let G be a complete multipartite graph with parts V1; V2; . . . ; Vsþt,

where vi¼ jVij is odd for 1 i t and is even for t þ 1 i s þ t. Let g be the

number of vertices of odd degree andm be the size of the largest part having vertices of odd degree. There exists a maximum4-cycle packing of 3G with leave L3where:

(i) ifg¼ 0 then jL3j 2 f0; 2; 3; 5g, and in particular if G ¼ Kð1; 1; nÞ then

L3¼ 3K2 when n is even,

3K2_ K1 when n is odd, and

(ii) ifg 1 then jL3j maxfg=2 þ 3; m þ 3g, except if G ¼ K3 or G¼ Kð1; nÞ, in

which cases L3¼ Eð3GÞ.

Remark. Note thatjL3j is completely determined by (i) and (ii), since if L0 is the

leave of any 4-cycle packing of 3G, then clearlyjL0_{j maxfg=2; mg and jL}0_{j jL} 3j

is divisible by 4.

Proof. (1) Sporadic Cases. If G¼ K3 or if G¼ Kð1; nÞ then 3G contains no

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If G¼ Kð1; 2n þ 1; 2Þ, then the union of a maximum 4-cycle packing of G (which has leave of size m¼ 2n þ 1) and a maximum 4-cycle packing of 2G (which has leave of size 2) produces a maximum 4-cycle packing of 3G with leave L3,

wherejL3j ¼ m þ 2.

If G¼ Kð1; 1; nÞ, then the union of a maximum 4-cycle packing of G (which has leave of size 1 if n is even and 3 if n is odd) and a maximum 4-cycle packing of 2G (which has leave of size 2) produces a maximum 4-cycle packing of 3G with leave L3, wherejL3j ¼ 3 if n is even and jL3j ¼ 5 if n is odd (so g ¼ 0).

Therefore, by Theorem 2.7, throughout the rest of the proof we can assume that for anyfu1; u2g 2 EðGÞ, if the number of edges in 2G is congruent to 2 (mod

4), then there exists a 4-cycle packing of 2G with leave ffu1; u2g; fu1; u2gg.

(2) t 0 or 1 (mod 4). In this case, when k ¼ 2 the leave is ;, and so a maximum 4-cycle packing in which the leave when k¼ 3 is exactly the same as when k ¼ 1, so the result follows from Theorem 1.1.

(3) t 2 or 3 (mod 4). Let M ¼ maxfm; g=2g. If jL1j M þ 1, we can add the

repeated edge leave L2, and the new leave L3satisﬁes (ii), so is a minimum leave as

required. So henceforth we assume thatjL1j 2 fM þ 2; M þ 3g.

We shall follow the order of the sections and adopt the notation used in [3]. The bipartite case

Both parts must have odd size since t 2 or 3 (mod 4). Let v1 v2. If v2 1 (mod

4) then jL1j ¼ v1¼ M. If v2 3 (mod 4), then we can use Lemma 3.1 since L1

contains D; L1is shown in Fig. 1 of [3]. So in this case, we havejL1j ¼ v1þ 2 and

jL3j ¼ jL1j 2 ¼ v1.

An odd number of parts, all of odd size

We must have t 3 (mod 4). We have two cases.

If t 3 (mod 8) then let ðV ; B1Þ be a maximum 4-cycle packing of G with leave

L1¼ K3, and letðV ; B2Þ be a maximum 4-cycle packing of 2G with leave L2¼ C2.

(Here C2denotes a pair of vertices joined by two edges.) ThenðV ; B1[ B2Þ is a

4-cycle packing of 3G with leave L3¼ K3[ C2 of size 5 as required.

If t 7 (mod 8), then let ðV ; B1Þ be a 4-cycle packing of G with minimum leave

the 5-cycle L1 ¼ ðu1; u4; u3; u2; u5Þ, and let ðV ; B2Þ be a 4-cycle packing of 2G with

leave L2¼ ffu1; u2g; fu1; u2gg. Then ðV ; B1[ B2[ fðu1; u2; u3; u4ÞgÞ is a 4-cycle

packing of 3G with leave L3¼ fðu1; u2; u5Þg ¼ K3, sojL3j ¼ 3.

An even number of parts, all of odd size

We refer the reader to Fig. 5 of [3], where the possibilities for the leave L1 of

ðV ; B1Þ are listed.

Case (i).

In this case,jL1j 2 fv1þ 2; v1þ 3g, so 2 or 3 copies of K2 in L1 do not contain a

vertex in V1; let fu1; u3g and fu2; u4g induce 2 such copies. Also, since

jL1j g=2 þ 2, in L1 there are 4 vertices in V1, say u5; u6; u7; and u8 that have a

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Q¼ ffu1; u3g; fu2; u4g; fa2; u5g; fa2; u6g; fa2; u7g; fa2; u8gg L1:

Also, the partition on the last line in page 115 in [3] ensures that

T1¼ fðu1; u5; u3; u6Þ; ðu2; u5; u4; u6Þ; ðu1; u7; u3; u8Þ; ðu2; u7; u4; u8Þg B1:

LetðV ; B2Þ be a 4-cycle packing of 2G with L2 ¼ ffu1; u2g; fu1; u2gg.

Also let T2¼ fðu3; u5; u4; u6Þ; ðu1; u2; u8; u3Þ; ðu1; u2; u4; u7Þ; ða2; u5; u2; u7Þ;

ða2; u6; u1; u8Þg. Then ðV ; ððB1n T1Þ [ B2[ T2ÞÞ is a 4-cycle packing of 3G with

leave L3¼ ðL1n QÞ [ ffui; uiþ4g j 1 i 4g. So jL3j ¼ jL1j 2.

Cases (ii) and (iv.3).

We can use Lemma 3.1 in this case, since L1contains D, and since B1contains a

4-cycle that uses 4 edges joining the vertices of degree 1 in D (see the ﬁrst sentence on page 115 of [3]).

Cases (iii) and (iv.1–2).

Here, L1contains two copies of K1;3(in Case (iii), this follows as described in Case

(i) since again jL1j g=2 þ 2, so in L1 there are 3 vertices of degree 1 in V1 that

have a common neighbour). Furthermore, each such copy of K1;3 contains a pair

of vertices of degree 1, say p1¼ fu1; u3g and p2¼ fu2; u4g respectively, such that

B1 contains the 4-cycleðu1; u2; u3; u4Þ. Now use Lemma 3.2.

Parts of both even and odd sizes

In this case we refer constantly to Section 5 of [3], adopting the notation deﬁned therein. In particular, S is a set of pairs of vertices, each pair containing two vertices from the same part; and if two such pairs contain vertices that occur in diﬀerent parts, then B1 contains the 4-cycle induced by these pairs. Moreover,

there exists a vertex z such that fz; ug 2 L1 for each vertex u in each pair in S.

As remarked in the second paragraph of Section 5 in [3], if t is even then the leave is identical to the leave formed if all vertices in parts of even size are deleted. So the result follows from the previous case (an even number of parts, all of odd size). So we can assume that t is odd, so therefore t 3 (mod 4).

Following the cases in [3], we obtain the following. s¼ 1 (page 120 of [3])

If t 3 (mod 8) then L1 contains the edges fz; u1g; fz; u2g; fz; u3g, and fu2; u3g,

where fz; u2; u3g O and fu1g E. So choose L2¼ ffu1; u2g; fu1; u2gg and let

B3¼ B1[ B2[ fða; u1; u2; u3Þg. Then jL3j ¼ jL1j 2, so ðV ; B3Þ provides the

re-quired 4-cycle packing.

If t 7 (mod 8) then we can simply use B2[ B2 since jL1j ¼ m þ 1, so then

jL3j ¼ m þ 3 as required.

s6¼ 1 and t 3 (mod 8)

IfjSj ¼ 0 then L1contains a copy of K3(see the last line on page 121 of [3]) with

vertex z joined to a vertex u12 E, so the result follows here by using exactly the

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If S contains two pairs from diﬀerent parts, then

jL1j ¼ g=2_v þ jSj 1 ifjSj 2 f2; 3g; and 1þ 1 ifjSj 4:

So we can assumejSj ¼ 3. Then by (ai) on page 121 of [3], all three pairs in S, say fu1; u5g; fu2; u4g, and fu3; u6g, occur in diﬀerent parts.

Let T1¼ fðu1; u2; u5; u4Þ; ðu1; u3; u5; u6Þg B1. And let

T2¼ fðu1; u2; z; u6Þ; ðu1; u2; u5; zÞ; ðz; u3; u5; u4Þg:

Let ðV ; B2Þ be a 4-cycle packing of 2G with leave L2¼ ffu1; u2g; fu1; u2gg. Then

ðV ; B3Þ is a 4-cycle packing of 3G where B3¼ ðB1n T1Þ [ B2[ T2 with

L3¼ ðL1n ffz; uig j 1 i 6gÞ [ fffu1; uig j 2 i 4g; fu5; u6gg:

Now suppose all pairs in S belong to one part. Then page 123 of [3] details two cases. One results in the leave described in equation (1) of [3] in which jL1j ¼ g=2 ¼ M so has already been considered (just take B1[ B2), and the other has

leave L1consisting of a copy of K3containing a vertex z joined to a vertex u12 E, so

this again reverts to the case where s¼ 1 and t 3 (mod 8), handled above. s6¼ 1 and t 7 (mod 8)

IfjSj ¼ 2 or jSj 4 and S contains two pairs from two diﬀerent parts (see page 124 of [3]), instead of applying Lemma 5.6 of [3] to the graph formed from K11 by

deleting the two disjoint edges joining vertices in p1 and p2, with vertex set

fzij 1 i 7g [ p1[ p2, to form B04, we supplement the set ~B1of 4-cycles deﬁned

so far, in the following way.

Let p1¼ fu1; u3g and p2¼ fu2; u4g. Let ðfzij 1 i 7g [ fu1; u2g; T1Þ be a

4-cycle system of K9, letððfzij 2 i 7g; fu3; u4gÞ; T2Þ be a 4-cycle system of K6;2,

and let T3 ¼ fðu1; u2; u3; u4Þg.

Let ðV ; B2Þ be a 4-cycle packing of 2G with leave L2¼ ffu1; u2g; fu1; u2gg.

ThenðV ; ~B1[ T1[ T2[ T3[ B2Þ is a 4-cycle packing of 3G with leave

L3¼ ðL0n ffz1; z2g; fz1; z3g; fz2; z3g; fu3; u4ggÞ [ ffz1; u3g; fz1; u4gg:

So jL3j ¼ jL0j 2. (See Fig. 1).

If jSj 6¼ 3 then let ðV ; B1Þ be the 4-cycle packing of G with leave L1 where

ffy1; z2g; fy2; z1g; fz1; z2gg L1:Then since we are assuming thatjL1j M þ 2, it

follows from the top of page 125 in [3] that jSj ¼ 1, say S ¼ fpg, and one of the sets S1; . . . ; Sl, say S1, contains at most one pair that is in the same part as p. Then

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we may let p¼ fy1; y2g and S1¼ fpi¼ fy2iþ1; y2iþ2g j 1 i 4g, named so that

possibly p and p1occur in the same part, possibly p2and p3occur in the same part,

but no other two pairs occur in the same part. If p and p1 occur in the same or

diﬀerent parts then let b1 ¼ ; or fðy1; y3; y2; y4Þg respectively, if p2 or p3 occur in

the same or diﬀerent parts then let b2¼ ; or fðy5; y7; y6; y8Þg respectively, and in

any case let b3¼ fðy1; y2iþ1; y2; y2iþ2Þ j 2 i 4g. On page 122 of [3], conditions

(1–2) show that b1[ b2[ b3 B1. Also in [3], the second last paragraph on page

121 shows that B1 B1is a maximum 4-cycle packing of Kð4; 4; 1Þ with partition

we can namefp1[ p4; p2[ p3;fz1gg. Also, let ðV ; B2Þ be a 4-cycle packing of 2G

with leave L2¼ ffy4; y8g; fy4; y8gg.

If B is a set of 4-cycles, then let EðBÞ denote the multiset of all edges occuring in the 4-cycles in B. Then one can check that, using the graph Hð1; . . . ; 12Þ deﬁned in 3.3, EðH ðz2; z1; y1; y2; . . . ; y10ÞÞ = EðB1[ b1[ b2[ b3Þ [ L1[ L2 (see Fig. 2).

Lemma 3.3 can now be used to form a 4-cycle packing T of Hðz2; z1; y1; y2; . . . ; y10Þ

with leave consisting of 5 independent edges.

Therefore, we can replace B1[ b1[ b2[ b3 in B1 with T to produce a 4-cycle

packing of 3G with leave L3, wherejL3j ¼ g=2.

Finally, supposejSj ¼ 3. By our assumptions, there are two pairs in V1 and a

third pair in another part. Instead of applying Lemma 5.6 to the graph formed from K11by deleting the two disjoint edges joining vertices in p1and p2, repeat the

process described at the start of this subsection, whenjSj ¼ 2 or jSj 4. (

4. The Cases k 4

We ﬁrst deal with one exceptional case.

If G¼ Kð1; 1; nÞ, then let V1 ¼ fu1g, V2 ¼ fu2g, and V3¼ fuij 3 i n þ 2g.

Clearly the edges joining u1to u2are in no 4-cycle in kG, so these edges must occur

in Lk. Simply take the union ofbk=2c copies of a maximum 4-cycle packing of 2G

with leave ffu1; u2g; fu1; u2gg together with, if k is odd, a maximum 4-cycle

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packing of G with leave eitherfu1; u2g (if n is even) or ffu1; u2g; fu2; u3g; fu3; u1gg

(if n is odd).

Next consider the case k¼ 4, for G 6¼ Kð1; 1; nÞ or Kð1; nÞ. We shall show that there is a maximum packing of 4G with 4-cycles having leave;. Despite the fact that the case k¼ 2 (in all but two exceptional cases) has leave either 2K2or;, in

order to obtain an empty leave when k¼ 4, we have some work to do!

First, note that if G contains no parts of size 1, then by [14], since k¼ 4, we can take all pairs of parts and pack each bipartite subgraph with 4-cycles with empty leave.

So now suppose that G contains at least one part of size 1.

If G contains one part of size 1, since G6¼ Kð1; nÞ, it must contain at least three parts. If G contains precisely three parts altogether, the other two parts are both of size greater than 1 (since G6¼ Kð1; 1; nÞ). We deal ﬁrst with some small cases, in the following seven lemmas. These small cases have three or four parts, and at least one part of size 1.

Lemma 4.1. There is a 4-cycle system of 4K(1, 2, 2).

Proof. Let V1 ¼ f1g, V2¼ f2; 3g and V3¼ f4; 5g. Take each of the following

4-cycles twice:

ð1; 2; 4; 3Þ; ð1; 2; 5; 3Þ; ð1; 4; 3; 5Þ; ð1; 4; 2; 5Þ:

(

Lemma 4.2. There is a 4-cycle system of 4K(1, 2, 3).

Proof. Let V1 ¼ f1g, V2¼ f2; 3g and V3¼ f4; 5; 6g. Then ðV1; V2; V3; BÞ is a 4-cycle

system, where

B¼ fð1; 2; 5; 3Þ; ð1; 2; 4; 3Þ; ð1; 2; 4; 3Þ; ð1; 2; 6; 3Þ; ð1; 5; 2; 6Þ; ð1; 5; 3; 6Þ; ð1; 4; 2; 5Þ; ð1; 4; 2; 6Þ; ð1; 4; 3; 5Þ; ð1; 4; 3; 6Þ; ð2; 5; 3; 6Þg:

( Lemma 4.3. There is a 4-cycle system of 4K(1, 3, 3).

Proof. Let V1¼ f1g, V2¼ fð0; 0Þ; ð1; 0Þ; ð2; 0Þg and V3¼ fð0; 1Þ; ð1; 1Þ; ð2; 1Þg.

For each i2 f0; 1; 2g let B ¼ ff1; ði; 0Þ; ði þ 2; 1Þ; ði þ 1; 0Þg; f1; ði; 0Þ; ði; 1Þ; ði þ 1; 0Þg; f1; ði; 1Þ; ði; 0Þ; ði þ 1; 1Þg; f1; ði; 1Þ; ði þ 1; 0Þ; ði þ 2; 1Þg; fði; 0Þ; ði; 1Þ; ði þ 1; 0Þ; ði þ 1; 1Þgg; reducing each sum modulo 3. Then ðV1; V2; V3; BÞ is the

required 4-cycle system.

( Lemma 4.4. There is a 4-cycle system of 4 K(1,1,1,1).

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Lemma 4.5. There is a 4-cycle system of 4K(1,1,1,2).

Proof. Let V1¼ f1g, V2¼ f2g, V3¼ f3g and V4¼ f4; 5g. Then ððV1; V2; V3; V4Þ; BÞ

is a 4-cycle system, where B¼ fð1; 2; 3; 4Þ; ð1; 2; 4; 3Þ; ð1; 2; 3; 5Þ; ð1; 2; 5; 3Þ; ð1; 4; 2; 3Þ; ð1; 4; 2; 5Þ; ð1; 4; 3; 5Þ; ð1; 5; 2; 3Þ; ð3; 4; 2; 5Þ:g ( Lemma 4.6. There is a 4-cycle system of 4K(1,1,2,2).

Proof. Let V1¼ f1g, V2¼ f2g, V3¼ f3; 4g and V4 ¼ f5; 6g. Then ððV1; V2; V3; V4Þ;

BÞ is a 4-cycle system, where

B¼ fð1; 2; 5; 3Þ; ð1; 2; 3; 5Þ; ð1; 2; 3; 6Þ; ð1; 2; 4; 5Þ; ð1; 3; 6; 4Þ; ð1; 3; 2; 5Þ; ð1; 3; 6; 4Þ; ð1; 4; 2; 6Þ; ð1; 4; 2; 6Þ; ð1; 5; 2; 6Þ; ð2; 3; 5; 4Þ; ð2; 5; 4; 6Þ; ð3; 5; 4; 6Þg:

( Lemma 4.7. There is a 4-cycle system of 4K(1,2,2,2).

Proof. LetjV1j ¼ 1 and jV2j ¼ jV3j ¼ jV4j ¼ 2. Take 4-cycle systems of 2Kð1; 2; 2Þ

on: fV1; V2; V3g, fV1; V2; V4g, fV1; V3; V4g, and 4-cycle systems of 2Kð2; 2Þ on:

fV2; V3g, fV2; V4g,fV3; V4g. (

We can now deal with 4-fold complete multipartite graphs.

Theorem 4.8. Let G be a complete multipartite graph. There exists a maximum 4-cycle packing of 4G with leave L where

(a) if G=Kð1,nÞ or Kð1; 1; 1Þ; then L=Eð4GÞ; (b) if n >1 and G=Kð1; 1;nÞ, then L ¼ 4K2, and

(c) L¼ ; otherwise.

Proof. Clearly Kð1; nÞ and Kð1; 1; 1Þ have no 4-cycles. Also, if G ¼ Kð1; 1; nÞ where n > 1 then the result follows by taking 2 copies of a 2-fold maximum 4-cycle packing.

Otherwise, ﬁrst suppose that G has n 4 parts. For 1 i 4 let Wi Viwith

jWij ¼ 1 or 2 if jVij 6¼ 2 or jVij ¼ 2 respectively. Let ððW1; W2; W3; W4Þ; T Þ be a

4-cycle system of 4KðjW1j; jW2j; jW3j; jW4jÞ (see Lemmas 4.4–4.7). Then

ðV ; T [ ðSn_i¼14BðVin Wi;ðSi1_j¼1WjÞ [ ðSn_j¼iþ1VjÞÞÞÞ is a 4-cycle system of 4G.

Suppose that G has three parts. Unless G¼ Kð1; 2; 3Þ or Kð1; 3; 3Þ, for 1 i 3 we can choose Wi Vi such that

(a) two of W1; W2 and W3 have size 2 and one has size 1, and

(b)jVin Wij 6¼ 2.

A 4-cycle system of 4G is provided in these two exceptional cases in Lemmas 4.2 and 4.3. So in each other case, let ððW1; W2; W3Þ; T Þ be a 4-cycle system of

KðjW1j; jW2j; jW3jÞ (see Lemma 4.1). Then ðV ; T [ ðS3i¼14BðVin Wi;ðSi1j¼1WjÞ[

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Corollary 4.9. Let G be a complete multipartite graph. There exists a 4-cycle system of 4G if and only if G6¼ Kð1; nÞ and G 6¼ Kð1; 1; nÞ.

For higher values of k¼ 4a þ b where 0 b < 4, we may simply combine a maximum packing with k¼ b (see Theorems 1.1, 2.7 and 3.4) together with a copies when k¼ 4 (see Theorem 4.8).

We may summarise this as follows.

Theorem 4.10. Let k 4 and let G be a complete multipartite graph. Let gðkÞ be the number of vertices of odd degree inkG, and let mðkÞ be the number of vertices in the largest part ofkG containing vertices of odd degree. There exists a maximum 4-cycle packing of kG with some leave Lk satisfying jLkj maxfgðkÞ= 2 þ 3; mðkÞ þ 3g,

except if

(i) G=Kð1;nÞ or Kð1; 1; 1Þ, in which case Lk¼ EðkGÞ;

(ii) G=Kð1; 1; nÞ with n > 1, in which case Lk¼ kK2_ K1

if n and k are odd; and kK2 otherwise;

(iii) gðkÞ ¼ 0, jEðkGÞj 1 (mod 4), and G 6¼ Kð1; 1; nÞ, in which case jLkj ¼ 5.

5. Conclusion

We now can summarise our work as follows.

Main Theorem. Let G be a complete multipartite graph. Let gðkÞ be the number of vertices of odd degree inkG, and let mðkÞ be the number of vertices in the largest part ofkG containing vertices of odd degree. There exists a maximum 4-cycle packing of kG with some leave Lk satisfyingjLkj ¼ l if and only if

(i) if G=ð1,nÞ or Kð1,1,1Þ, then Lk¼ EðkGÞ,

(ii) if G=Kð1,1,nÞ and n > 1 then Lk¼

kK2_ K1 if n and k are odd; and

kK2 otherwise;

(iii) ifgðkÞ ¼ 0, jEðkGÞj 1 (mod 4), and G 6¼ Kð1; 1; nÞ, then jLkj ¼ 5,

(iv) ifgðkÞ ¼ 0, jEðkGÞj 2 (mod 4), and k ¼ 1, then jLkj ¼ 6, and otherwise

(v) l is the unique integer satisfying

(1) maxfgðkÞ=2; mðkÞg l maxfgðkÞ=2 þ 3; mðkÞ þ 3g, and ð2Þ 4 divides jEðkGÞj l.

Proof. The necessity of conditions (i)–(v) was dealt with in Lemma 1.2. The suﬃciency follows from Theorem 1.1 for k¼ 1, Theorem 2.7 for k ¼ 2, Theorem

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Clearly this research raises several interesting related questions. The method of attack used here suggests that ﬁnding all possible leaves of maximum packings may well be possible, although it is probably a lot of work, and perhaps not so easy to display. Moreover, the the minimum covering problem is a natural fol-lowup; it is likely that such problems can be attacked by using the results in this paper.

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Received: August 26, 2003