Packing complete multipartite graphs with 4-cycles

Packing Complete Multipartite

Graphs with 4-cycles

Elizabeth J. Billington,1 _{Hung-Lin Fu,}2_{C.A. Rodger}3

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, 120 Math Annex,} Auburn University, Auburn, Alabama 36849±5307, U.S.A.

E-mail: rodgec1@mail.auburn.edu

Received November 29, 1999; acctepted April 20, 2000

Abstract: In this paper we completely solve the problem of ®nding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.# 2001 John Wiley & Sons, Inc. J Combin Designs 9: 107±127, 2001

Keywords: cycle packing; multipartite; 4-cycles

1. INTRODUCTION AND PRELIMINARIES

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

leave of a k-cycle packing of G is the set of edges of G that occur in no k-cycle in C; sometimes we also refer to the subgraph induced by these edges as the leave. A k-cycle system of G is a k-cycle packing of G for which the leave is empty. We refer to the leave of a maximum k-cycle packing as a minimum leave. Also, let K…v1; v2; . . . ; vn† denote the complete multipartite graph with vertex set

V1[ V2[


[ Vn and edge set E, where jVij ˆ vi and E consists of all edges

between vertices in Vi and Vj, i 6ˆ j; there are no edges between two vertices in the

same set Vi.

Contract grant sponsor: Australian Research Council; Contract grant number: A69701550; Contract grant sponsor: NSC; Contract grant number: 88-2115-M-009-013; Contract grant sponsor: NSF; Contract grant number: DMS-9531722; Contract grant sponsor: ONR; Contract grant number: N00014-97-1-1067.

# 2001 John Wiley & Sons, Inc.

In recent years, various edge-disjoint decompositions of complete graphs and complete multipartite graphs into cycles have been investigated; see for example [6] and [2]. Moreover, maximum packings and minimum coverings of complete graphs by k-cycles for various k have also been considered; see [8], [7], and [5] for 4-, 5- and 6-cycles respectively, for instance.

The problem of partitioning the edges of a complete multipartite graph into 3-cycles has also been considered, and is proving to be an extremely dif®cult problem to solve. For example, one paper deals with the particular case where all parts have the same size, except possibly for one part [3]. In contrast to this, here we completely solve the problem of ®nding a maximum 4-cycle packing of K…v1; v2; . . . ; vn† (see

Theorem 6.1). This generalizes the result of Cavenagh and Billington [2] which characterizes the complete multipartite graphs for which there exists a 4-cycle system.

This problem has already been solved for complete graphs; that is, when v1 ˆ v2 ˆ


ˆ vn ˆ 1. For convenience, in Table I we list the minimum leaves in

this case (see [8], and also [4]). In the following, F denotes a 1-factor of the complete graph Kn when n is even, B denotes a bowtie, that is, two triangles K3 having one

common vertex, and Ci denotes a cycle of length i.

Remark 1.1. It is also possible (and will be useful in a later section) to obtain a packing of Kn with 4-cycles, having leave Ki, when n  i (mod 8), for i ˆ 1; 3; 5; 7.

(Clearly, this is not a maximum packing when i ˆ 5 or 7, but by replacing the Kiby its

maximum packing, we can obtain a maximum packing of Kn, containing a maximum

packing of Ki, i ˆ 5 or 7. See the inductive construction described in [4].)

One straightforward result (which is easily seen to hold for 4-cycles) follows from Sotteau [9]. This guarantees the existence of a decomposition of any complete bipartite graph into 4-cycles if and only if the two parts each have even size. We shall use this frequently in the following. This result also means that in any complete multipartite graph which has all parts of even size, there is a decomposition into 4-cycles with empty leave. We shall refer to this as the ``all parts even'' condition (*), and henceforth assume that at least one part has odd size.

The complement of a graph G is denoted here by G. If two graphs G and H are vertex disjoint, then the join G _ H is formed from G [ H by joining each vertex in G to each vertex in H. For any other graph theoretic de®nitions, see [10].

2. THE BIPARTITE CASE

Let K…v1; v2† be a complete bipartite graph with vertex partition fV1; V2g where

jVij ˆ vi, i ˆ 1; 2. If both v1 and v2 are even, condition (*) ensures there exists a

4-cycle system of K…v1; v2†.

TABLE I. Minimum leaves in 4-cycle packings of Kn

Kn, n mod 8: 1 2 3 4 5 6 7 0

If v1is odd and v2 is even, then in any minimum leave each vertex in V2 has odd

degree. Pick any vertex x1 in V1, and let R denote the star centered at x1, with v2

edges. This is our leave. What remains is the graph K…v1; v2† n E…R†, which is

isomorphic to K…v1ÿ 1; v2†, a complete bipartite graph with both parts of even size.

So a 4-cycle decomposition of K…v1; v2† n E…R† follows from (*).

Now suppose that v1and v2are odd, with v1 v2. In this case any minimum leave

must be a spanning subgraph, with every vertex of odd degree. So, certainly the minimum number of edges in the leave is at least v1. Thus, if v2 3 (mod 4), any

minimum leave must contain at least v1‡ 2 edges, in order that the number of

remaining edges is 0 (mod 4). Therefore, the graph induced by the minimum leaves can be F1 or F2, according as v2 1 or 3 (mod 4), where F1 and F2 are given in

Figure 1 (providing the remaining edges can be partitioned into 4-cycles).

In Figure 1, we conveniently group vertices in the component of F1 or F2

con-taining more than one edge into pairs p1;i as shown, together with the special pair

a1; a2. This concept of paired vertices will also be important in Section 4 below.

Note that the number of components in both F1 and F2 is 1 (mod 4). In order to

describe a convenient 4-cycle decomposition of K…v1; v2† n Fi, i ˆ 1; 2, we need the

following lemma. (See also Lemma 6 of [1]; we include a brief proof below for completeness.)

Lemma 2.1. The complete bipartite graph K…4m ‡ 1; 4m ‡ 1† minus a perfect matching F has a decomposition into 4-cycles.

Proof. First, when m ˆ 1, a decomposition of K5;5n F with vertex set

f0; 1; 2; 3; 4g [ f00_{; 1}0_{; 2}0_{; 3}0_{; 4}0_{g into 4-cycles is given cyclically by …0; 1}0_{; 4; 3}0_†

(mod 5), where F is ffi; i0_{g j 0  i  4g.}

Now, K…4m ‡ 1; 4m ‡ 1† n F is essentially m copies of K5;5n F (with vertices 0

and 00_{in each copy, and with f0; 0}0_{g 2 F), together with m…m ÿ 1† copies of K} 4;4. So

the result follows. &

Returning to K…v1; v2† n E…Fi†, we can now apply Lemma 2.1 together with (*),

and easily decompose K…v1; v2† n E…Fi† into 4-cycles. Thus we have proved:

Lemma 2.2. The complete bipartite graph K…v1; v2† can be decomposed into

4-cycles with leave L, where L is as follows:

K…v1; v2† leave L

v1  v2  0 (mod 2) ;

v1ÿ 1  v2  0 (mod 2) R, star with v2 edges

v1  v2  1 (mod 2), v1 v2  1 (mod 4) F1 (see Figure 1)

v1  v2  1 (mod 2), v1 v2  3 (mod 4) F2 (see Figure 1) FIG. 1.

In the case v2  3 (mod 4), for 1  i  x1ÿ 1; the decomposition includes the

4-cycle with vertex set p1;x1[ p1;i.

3. AN ODD NUMBER OF PARTS, ALL OF ODD SIZE

In this case the vertices are in parts Vi, 1  i  n, with n odd and viodd. Let wi 2 Vi,

for 1  i  n, and let Vin fwig be denoted by Qi. Then we may take a maximum

packing as follows, with leave being exactly the same as the leave for a maximum packing of Kn with 4-cycles (see Table I).

First, on the set fwi j 1  i  ng, place a maximum packing of Kn with 4-cycles.

Then use (*) to take a 4-cycle decomposition of the following complete bipartite graphs Fi and Hij, for 1  i; j  n, i 6ˆ j. The graph Fi has vertex partition

ffwjj 1  j  n; j 6ˆ ig; Qig, while Hij, i < j, has vertex partition fQi; Qjg.

Now each edge of the complete multipartite graph with an odd number of odd parts is used either in the leave or in a 4-cycle. Furthermore, the leave has at most six edges, so since it must be simple, the leave is a minimum leave.

We summarize this section as follows.

Lemma 3.1. A maximum packing with 4-cycles of a complete multipartite graph with n parts, where all parts have odd size and where n is odd, has minimum leave exactly the same as that in a maximum packing of Kn, namely: ;; K3; B; C6 or 2K3;

C5, according as n  1; 3; 5 or 7 (mod 8).

Remark 3.2. We may also take a packing of a complete multipartite graph with n odd parts, where n is odd, having leave as described in Remark 1.1, namely: ;, K3,

K5, K7according as n  1, 3, 5 or 7 (mod 8). (Of course, this is not a minimum leave

when n  5 or 7 (mod 8), but this type of leave will be useful later.)

4. AN EVEN NUMBER OF PARTS, ALL OF ODD SIZE

In this section we deal with one of the two dif®cult cases. We begin with some preliminary results giving 4-cycle decompositions of particular graphs which arise later.

For 1  i  4, let pibe a set of two of non-adjacent vertices. Let H1…b1; p1; b2; p2;

b3; p3; b4; p4† denote the graph with vertex set fbij 1  i  4g [ …S4iˆ1pi† and edge

set consisting of the eight edges joining bi to vertices in pi, for 1  i  4, together

with the edges of a K4;4 with bipartition p1[ p2 and p3[ p4 (see Fig. 2). Here

possibly b1 ˆ b2, and possibly b3ˆ b4. This graph H1 contains 24 edges, and the

degrees of the 8 vertices inS4_iˆ1 pi are all odd.

Lemma 4.1. The graph H1…b1; p1; b2; p2; b3; p3; b4; p4† has a 4-cycle packing with

the leave perfectly matching between p1[ p2 and p3[ p4.

Proof. Letting piˆ fci;1; ci;2g, 1  i  4, the leave is the four edges fc1;1; c4;2g;

fc1;2; c3;2g; fc2;1; c4;1g; fc2;2; c3;1g, and the 4-cycles are …b1; c1;1; c3;1; c1;2†; …b2; c2;1;

c4;2; c2;2†; …b3; c3;1; c2;1; c3;2†; …b4; c4;1; c1;2; c4;2†; …c1;1; c3;2; c2;2; c4;1†. (see Fig. 2.). &

We now de®ne the graph H2…W1; W2; W3; b1; b2; c1; c2; c3; c4† as follows (see Fig.

3). The sets W1 and W2 each consists of four different vertices, with a copy of K4;4

joining them. Vertex bi is joined by an edge to each of the four vertices in Wi,

i ˆ 1; 2. The set W3 consists of eight independent vertices, paired as p1; p2; p3; p4, so

that ci is joined by edges to the two vertices in pi, 1  i  4. Finally, H2 contains a

copy of K8;8 with bipartition W1[ W2 and W3. Note that possibly the vertices

c1; c2; c3; c4 are not all distinct (see Fig. 3).

Lemma 4.2. The graph H2…W1; W2; W3; b1; b2; c1; c2; c3; c4† has a 4-cycle packing

with the leave being a perfect matching between W1[ W2 and W3.

FIG. 2. H1…b1; p1; b2; p2; b3; p3; b4; p4† with its maximum 4-cycle decomposition.

Proof. The graph H2 is made up of: H1…c1; p1; c2; p2; b1; p5; b1; p6† (where

W1ˆ p5[ p6†; H1…c3; p3; c4; p4; b2; p7; b2; p8† (where W2ˆ p7[ p8†; and three

copies of K4;4, one from W1 to W2, one from p1[ p2 to W2, and one from p3[ p4

to W1. Therefore, the result follows from Lemma 4.1 and the fact that K4;4is trivially

decomposable into 4-cycles. &

The next lemma is similiar to the previous two in ¯avor, and is needed subse-quently in one particular case. We de®ne the graph H3…W1; W2; W3; W4; b1; b2; b3; b4†

on 20 vertices as follows (see Fig. 4). The sets W1 ˆ fw1;ij 1  i  4g and

W2ˆ fw2;ij 1  i  4g each consists of four independent vertices, with a copy of

K4;4joining them. Vertex b4 is joined to the four vertices in W1, vertex b3is joined to

the ®ve vertices in W2[ fb4g. The set W4 consists of two independent vertices, each

joined to b4. The set W3 consists of six independent vertices, w3;i, 1  i  6, all

joined to b2, and b2 is joined to b1. Finally, H3contains copies of K6;4, K6;4, and K2;4,

with bipartitions fW3; W1g, fW3; W2g, and fW4; W2g, respectively.

Lemma 4.3. The graph H3…W1; W2; W3; W4; b1; b2; b3; b4† can be decomposed into

4-cycles with the leave being a 1-factor consisting of fb1; b2g, fb3; b4g, four edges

between fw3;1; w3;2; w3;3; w3;4g and W1, and four edges between fw3;5; w3;6g [ W4

and W2.

Proof. The graph H3 is made up of: H1…b2; fw3;1; w3;2g; b2; fw3;3; w3;4g;

b4; fw1;1; w1;2g; b4; fw1;3; w1;4g†; H1…b2; fw3;5; w3;6g; b4; W4; b3; fw2;1; w2;2g; b3;

fw2;3; w2;4g†; a copy of K8;4 joining W1[ fw3;1; w3;2; w3;3; w3;4g to W2; a copy of

K2;4 joining fw3;5; w3;6g to W1; and the two edges b1b2 and b3b4. Thus the

decomposition into 4-cycles follows from Lemma 4.1 and condition (*). & FIG. 4 The graph H3…W1; W2; W3; W4; b1; b2; b3; b4†.

Lemma 4.4. The graph K9_ K9 has a 4-cycle packing with the leave being a

perfect matching of nine edges.

Proof. The graph K9 can be packed with 4-cycles (with empty leave), since 9  1

(mod 8). Also, K9;9can be decomposed into one perfect matching and a collection of

4-cycles (Lemma 2.1). So the result follows. &

Let H4…b1; b2; . . . ; b7; b8; . . . ; b11; b12; b13; b14† denote the graph K7_ …K4_ K3†,

where V…K7† ˆ fb1; . . . ; b7g, V…K4† ˆ fb8; . . . ; b11g; and V…K3† ˆ fb12; b13; b14g.

Lemma 4.5. The graph H4…b1; . . . ; b7; b8; . . . ; b11; b12; b13; b14† has a 4-cycle

packing, with the leave being a perfect matching of seven edges.

Proof. Begin with a 4-cycle packing of K7;7, with bipartition of the vertices being

ffb1; . . . ; b7g, fb8; . . . ; b14gg, so that the leave is the set of edges ffb1; b8g;

fb2; b9g; fb3; b10g; fb4; b11g; fb5; b12g; fb5; b13g; fb5; b14g; fb6; b12g; fb7; b12gg, and

so that …b6; b13; b7; b14† is one of the 4-cycles (see Lemma 2.2). Remove the

4-cycle …b6; b13; b7; b14†; the leave from K7;7 now consists of seven copies of K2

together with one 6-cycle c1 ˆ …b5; b13; b7; b12; b6; b14†.

We also have a partition of E…K7n K3†, with V…K7n K3† ˆ fb8; b9; . . . ; b14g and

V…K3† ˆ fb12; b13; b14g, which induces three 4-cycles and one 6-cycle:

f…b8; b9; b12; b10†; …b9; b10; b11; b14†; …b8; b11; b9; b13†; and c2 ˆ …b8; b12; b11; b13;

b10; b14†g.

The edges in the two 6-cycles c1 and c2 together form three 4-cycles:

…b5; b13; b10; b14†, …b6; b14; b8; b12†, …b7; b12; b11; b13†. The only remaining leave is

now ffbi; bi‡7g j 1  i  7g. &

We are now ready to prove our main result in this section.

Theorem 4.6. Suppose G is a complete multipartite graph with 2z parts V1; . . . ; V2z,

where jVij ˆ vi is odd for 1  i  2z: Also, let  be the number of vertices in the

largest part, and let  ˆP2z_iˆ1vi. There exists a maximum 4-cycle packing in which

the leave L satis®es either (A) jLj ₂‡ 3, or (B) jLj   ‡ 3.

(Each leave constructed here induces one of the graphs in Fig. 5.)

Proof. Since every vertex in G has odd degree, we ®rst point out that any leave L will be a spanning subgraph with all vertices of odd degree; so, clearly L is a minimum leave if jLj  maxf=2 ‡ 3;  ‡ 3g.

We pair the parts of G, V2iÿ1with V2ifor 1  i  z; for convenience we label them

v2iÿ1 v2i for each pair of parts V2iÿ1; V2i.

For 1  i  z; apply Lemma 2.2 to the bipartite graph with vertex partition V2iÿ1; V2i to obtain a maximum 4-cycle packing, Bi, and let Gi be the graph

containing the edges in the leave of Bi (the 4-cycles in Bi might not be part of our

®nal set). Note that Gicontains v2iÿ1edges if v2i 1 (mod 4), and v2iÿ1‡ 2 edges if

v2i 3 (mod 4). For 1  i  z, Gi consists of "i copies of K2 (where necessarily

(mod 4), then Zi K2; otherwise, Ziis the star Riif v2i 1 (mod 4) and v2iÿ1 > v2i,

and Zi Diif v2i 3 (mod 4) (see Fig. 6). We now introduce the notation used for

certain vertices in Figure 6. In the star Ri, pair off all but one of the vertices of degree

1 in V2iÿ1 into sets pi;1; pi;2; . . . ; pi;xi, and in Dipair off all of the vertices of degree 1

in V2iÿ1into sets pi;1; . . . ; pi;xiÿ1and let pi;xibe the pair of vertices of degree 1 in V2iin

Di. In any case, let a2iÿ1 and a2i in Zi be the unpaired vertices in Zi\ V2iÿ1 and

Zi\ V2i, respectively. Note that in each leave the two vertices in each pair pi;j have a FIG. 5. Leaves: 2z parts, all of odd size.

common neighbor. Note also that if pi; j ˆ fx1; y1g and pi;xi ˆ fx2; y2g occur in

different parts in Zi(see Diin Fig. 6), then by Lemma 2.2 the 4-cycle …x1; x2; y1; y2† is

in Bi.

Now we concentrate on the pairs pi; j, 1  i  z, 1  j  xi. We partition as many

of these pairs as possible into sets S1; S2; . . . ; Sy of size 4, with the property that for

1  k  2z and 1  l  y, the set Slcontains at most two pairs from each part Vk. Let

S denote the set of remaining pairs which do not occur in Sy_lˆ1Sl. Then S must

satisfy:

(a) jSj  3, or

(b) jSj  4 and all pairs in S, except possibly one, belong to the same part, say V1,

in G1.

In both cases (a) and (b), if Slˆ fpi1;m1; pi2;m2; pi3;m3; pi4;m4g, then we apply Lemma

4.1 to the graph H1…b1; pi1;m1; b2; pi2;m2; b3; pi3;m3; b4; pi4;m4† where bt ˆ a2itÿ1 or a2it

according as pit;mt is in V2it or V2itÿ1 for 1  t  4 (so possibly b1ˆ b2, possibly

b3 ˆ b4, possibly i1 ˆ i2, and possibly i3 ˆ i4). Let B1;ldenote the set of ®ve 4-cycles

obtained by applying Lemma 4.1 to Slin this way, and let Lldenote the four edges in

the leave. These are placed in our ®nal set of 4-cycles, B, and our ®nal leave L, respectively.

Now for convenience, let Ei denote the set of i copies of K2 in the leave Gin Zi,

1  i  z. (Henceforth we also think of an edge such as fa2iÿ1; a2ig as a 2-element set

of vertices.) Then we partition all the vertices in V into 2-element subsets: let P ˆ fe 2 Ei; pi; j; fa2iÿ1; a2ig j 1  i  z; 1  j  xig:

In case (a), for every two distinct pairs fq1; q2g and fq3; q4g in P that are not both in

Sl for 1  l  y, and not both contained in the same V…Gi†, 1  i  z, we take the

4-cycle …q1; q3; q2; q4† and place it in B. In case (a) we complete forming B by adding

all the 4-cycles inSziˆ1Bi, except those 4-cycles which have been used in taking care

of the pairs in the Sl. Let G0iˆ …Gin Zi†, and let L0i be formed from E…G0i† by adding

fa2iÿ1; a2ig and any edges in Githat are incident with vertices in pairs in S. Then the

leave of B is L ˆ …Sy_lˆ1Ll† [ …Sziˆ1L0i†, and jLj ˆ =2 ‡ jSj  =2 ‡ 3.

The minimum leave is given in Figure 5; we comment further on this at the end of the proof.

Case (b) remains. By choosing y to be maximal, we can assume (for 1  l  y† that each Slcontains exactly two pairs in V1in G1. (For, if Slcontains at most one pair

from V1then we can replace any pair in Slthat is not in V1with a pair from S that is in

V1.) So we can assume that V1 is the largest part, and thus v1 ˆ .

Recall that for 1  i  z, Eiˆ E…Gi† n E…Zi†, "iˆ jEij, and "i 0 (mod 4). Let

 ˆ min jSj₄   ; Xz iˆ2 "i 4 ( ) :

(Thus  is the minimum of the number of disjoint sets of four distinct pairs that are in S, and the number of disjoint sets of four copies of K2 that are inSziˆ2Gin Zi.)

Now select  pairwise disjoint sets T1; . . . ; T, where each Tj contains four edges

from Ei, for some i, 2  i  z, as well as four pairs from S.

Consider Tj. Let e1; . . . ; e4 be the four edges from Eiin Tj, so each joins a vertex

in part V2iÿ1 to a vertex in V2i, for some i  2. Also, let p1; j1; p1; j2; p1; j3; pk; j4 be the

four pairs from S in Tj. Here possibly k ˆ 1 with pk; j4 in V1 or in V2, or k > 1,

in which case without loss of generality we say pk; j4 is in V3. So the possibilities

for Tj are:

T(i) 4 pairs in Tj all from V1, 4 edges e1; . . . ; e4 all from Gi, for some i,

2  i  z.

T(ii) 3 pairs in Tjfrom V1, one pair from V2, 4 edges e1; . . . ; e4from Gi, for some

i, 2  i  z.

T(iii) 3 pairs in Tjfrom V1, one pair from V3, 4 edges e1; . . . ; e4from Gi, for some

i, 3  i  z.

T(iv) 3 pairs in Tjfrom V1, one pair from V3, 4 edges e1; . . . ; e4from G2(based on

V3[ V4).

In cases T(i), T(ii), and T(iii) above, we apply Lemma 4.2 to the graph H2…W1; W2; W3; b1; b2; c1; c2; c3; c4† where: c1ˆ c2ˆ c3ˆ a2; in cases T(i), T(ii),

and T(iii), c4 ˆ a2; a1; and a4; respectively; W1 ˆ …S4jˆ1ej† \ V2iÿ1 and W2ˆ

…S4_jˆ1ej† \ V2i (regarding ej as a set of two vertices); W3ˆS3lˆ1p1; jl [ pk; j4; and

b1 ˆ a2iand b2 ˆ a2iÿ1. If k 6ˆ 1 then for 1  l  3 add the 4-cycle between p1; jl and

pk; j4 to the set of 4-cycles that arises in this way from Tj, and call the resulting set

B2; j. Let B2; j  B, and place the leave from Lemma 4.2 into L.

In case T(iv) above, we apply Lemma 4.3 to the graph H3…W1; W2; W3; W4;

b1; b2; b3; b4† where W1ˆ …S4jˆ1ej† \ V3, W2ˆ …S4jˆ1ej† \ V4, W3 ˆS3lˆ1p1;jl,

p2;j4 and p1;jl to the set of 4-cycles obtained from Lemma 4.3 to form B2;j. Let

B2;j B, and place the leave from Lemma 4.3 into L.

Let S0_{ˆ S n fp}

i;lj pi;l2 Tj for some j; 1  j  g, and let A ˆ ffa2iÿ1; a2ig j

2  i  zg. Choose  pairwise disjoint sets U1; . . . ; U with  as large as possible so

that for 1  j  

U(i) Ujcontains 4 pairs in S0, each of which is a subset of V1, and 4 edges in A,

or

U(ii) Ujcontains 4 pairs in S0, three of which are subsets of V1and one is a subset

of V2, and 2 edges in A, or

U(iii) Ujcontains 4 pairs in S0, three of which are subsets of V1and one is a subset

of Vj with j  3 (say j ˆ 3), and 2 edges in A, one of which is fa3; a4g.

Notice that since S contains at most one pair that is not a subset of V1 (in case …b†),

we have at most one occurrence of cases U(ii) and U(iii); in such a case we can assume that the edges in A are fa3; a4g and fa5; a6g.

If Uj ˆ fp1;jl; fa3; a4g; fa5; a6g j 1  l  4g is formed in case U(ii), then let

p1;jl ˆ fp11;jl; p 2

1;jlg for 1  l  4 with p1;j4  V2. Let B3;j be the set of 4-cycles

obtained by applying Lemma 4.5 to the graph H4…a1; p11; j1; p 2 1; j1; p 1 1; j2; p 2 1; j2; p 1 1; j3; p 2 1; j3; a3; a4; a5; a6; a2; p11; j4; p 2

1; j4†, and let L3;j be the leave.

If Uj ˆ fp1;jl; p2;j4; fa3; a4g; fa5; a6g j 1  l  3g is formed in case U(iii) then let

pi;jl ˆ fp1i;jl; p 2

i;jlg for 1  j  4 and i 2 f1; 2g, with p2;j4  V3. Let B3;jbe the set of

4-cycles obtained by applying Lemma 4.5 to the graph H4…a1; p11; j1; p 2 1; j1; p 1 1; j2; p 2 1; j2; p1 1; j3; p 2 1; j3; a3; a4; a5; a6; a2; p 1 2; j4; p 2

2;j4†, and let L3;j be the leave.

Finally, if Ujˆ fp1; jl; elj 1  l  4g then let B3;j be the set of 4-cycles obtained

by applying Lemma 4.4 to the graph K9_ K9 with V…K9† ˆ fa1g [ …S4jˆ1p1; j† and

V…K9† ˆ a2[ …S4jˆ1ej†, and let L3;j be the leave.

For 1  j   let B3;j B and let L3;j L.

Now let V0

j; 1  j  2z, be the subset of Vj containing all vertices that are not

incident with an edge inS_jˆ1Tj. Also, let B0i be the set of 4-cycles in a maximum

packing of the complete bipartite graph V0

2iÿ1[ V2i0, chosen so that the leave G0i

satis®es G0

i Gi; this is possible since jVjj  jVj0j (mod 4) for 1  j  2z. Let B0i  B.

All that remains is to include those 4-cycles arising from decompositions of various complete bipartite subgraphs, all of whose parts now have even size (so condition (*) applies). First, for 2  i  z, we place in B the 4-cycles of a 4-cycle system of each of the two complete bipartite graphs, one with bipartition V0

2iÿ1n

fa2iÿ1g and V2in V2i0, the other with bipartition V2i0 n fa2ig and V2iÿ1n V2iÿ10 .

Secondly, recall that V is partitioned into 2-element subsets, P ˆ fe 2 Ei; pi;j;

fa2iÿ1; a2ig j 1  j  xi; 1  i  zg. Let fq1; q2g, fq3; q4g be any two of the

2-element subsets in P satisfying:

1. ffq1; q2g; fq3; q4gg 6 Sl, 1  l  y,

2. ffq1; q2g; fq3; q4gg 6 Ti, 1  i  ,

3. ffq1; q2g; fq3; q4gg 6 Ui, 1  i  , and

4. fq1; q2; q3; q4g 6 V…Gi† for 1  i  z,

(again, regarding the edges in Sl, Ti, and Uias 2-element subsets). Then we place the

Below we comment on the leaves described in Figure 5. Recall that since every vertex must have odd degree in the leave, it is clear that if

(a) the number of edges is less than₂‡ 4, or (b) the number of edges is less than  ‡ 4,

then the leave is a minimum. In the case (a) above (when jSj  3), any of the leaves (i)±(iv) can arise, while in case (b) above (when jSj  4), one of the leaves (i)±(iii) arises. Moreover, cases (i), (ii), and (iii) of Figure 5 satisfy (B), while case (iv) satis®es (A). Thus in all cases the resulting leave is a minimum, and we have achieved our desired maximum packing.

This concludes the proof of the theorem. &

5. PARTS OF BOTH EVEN AND ODD SIZES

Here we deal with the ®nal case, which in retrospect is probably the most dif®cult case, but easier to read!

First, note that if the number of parts, say t, of odd size is even (and possibly zero), then this case essentially reduces to that in Section 4 above as the following shows. Let V1; . . . ; Vsbe the parts of even size, and let Vs‡1; . . . ; Vs‡tbe the parts of odd size.

We use condition (*) on the bipartite graphs with bipartition fVi; Vjg for

1  i < j  s, and also on the complete bipartite graph with bipartition fV1[ V2[


[ Vs; Vs‡1[ Vs‡2[


[ Vs‡tg. Finally, Theorem 4.6 above deals

with the complete multipartite graph that remains, on the parts Vs‡1; . . . ; Vs‡t. The

resulting leave is clearly a minimum since it satis®es (A) or (B), where  is now the number of vertices of odd degree, and  is the size of the largest part.

However, if the number of odd parts is odd, the above simple approach needs modi®cation. First we give some useful lemmas.

Lemma 5.1. The graph K…4; 4; 1† can be decomposed into ®ve 4-cycles and four independent edges.

Proof. Let the vertex set be partitioned as ffa1; a2; a3; a4g; fb1; b2; b3; b4g; fzgg.

Then the decomposition is …z; a1; b3; a2†; …z; a3; b2; a4†; …z; b1; a4; b3†; …z; b2; a1; b4†;

…a2; b1; a3; b4†, and the edges in ffai; big j 1  i  4g. &

Lemma 5.2. The graph K7 minus one edge can be decomposed into four 4-cycles

and a path of length 4.

Proof. We may pack K7 with 4-cycles with minimum leave one 5-cycle (see Table

I). Let the removed edge be from this 5-cycle, and the result follows. &

Lemma 5.3. The graph K9 minus one edge can be decomposed into eight 4-cycles

and a path of length 3.

Proof. A maximum packing of K9 with 4-cycles has empty leave (see Table I). So

removing one edge from one 4-cycle produces the required leave of a path of length 3.

Lemma 5.4. Let K7 be de®ned on the vertex set fa1; . . . ; a7g. Then K7 minus two

vertex-disjoint edges fa1; a2g and fa3; a4g can be decomposed into four 4-cycles with

leave ffa1; a3g; fa2; a5g; fa4; a5gg.

Proof. Take the 4-cycles …a2; a3; a5; a6†; …a1; a4; a7; a5†; …a1; a6; a3; a7†; …a2; a4;

a6; a7†; this leaves edges fa1; a3g; fa2; a5g and fa4; a5g. &

Lemma 5.5. The graph K9 minus two vertex-disjoint edges fa1; a2g and fa3; a4g

can be decomposed into eight 4-cycles with leave ffa1; a3g; fa2; a4gg.

Proof. Take a 4-cycle decomposition of K9 (with empty leave) containing the

4-cycle …a1; a2; a4; a3†. Then removal of the edges fa1; a2g and fa3; a4g yields the

result. &

Lemma 5.6. The graph K11 minus two vertex-disjoint edges fa1; a2g and fa3; a4g

can be decomposed into twelve 4-cycles and leave the ®ve edges e1ˆ fa1; a3g;

e2ˆ fa2; a4g and three edges that induce a copy of K3 that is vertex-disjoint from

e1[ e2.

Proof. Start with a 4-cycle packing of K11, which contains thirteen 4-cycles and has

leave K3(see Table I). We choose one 4-cycle vertex-disjoint from the K3leave, and

label it …a1; a2; a4; a3† so that removal of the two disjoint edges leaves the opposite

edges, fa1; a3g; fa2; a4g. We can certainly ®nd such a disjoint 4-cycle, since the

vertices in the K3leave are each in four 4-cycles, whereas the packing of K11contains

thirteen 4-cycles, and 13 > 4  3. &

Lemma 5.7. There exists a packing of K…4; 4; 2; 1; 1; 1† with 4-cycles in which the leave L is a matching of ®ve edges that saturates the ten vertices of odd degree. Proof. Let v1ˆ v2 ˆ 4; v3 ˆ 2, and v4 ˆ v5 ˆ v6 ˆ 1. Let Vi ˆ fwi;j j 1  j  vig

for 1  i  6. Let B1 be a set of ®ve 4-cycles that packs K5;5 with bipartition

fV1[ fw4;1g; V2[ fw5;1gg in which the leave is ffw1;j; w2;jg; fw1;4; w5;1g;

fw2;4; w4;1g j 1  j  3g. (Only some of this leave is in the ®nal leave.) Let B2 be

the set of 4-cycles in a 4-cycle system of K2;6 with bipartition fV3; fwi;jj 1  i  2;

1  j  3gg. Let B3ˆ f…wi;1; w6;1; wi;2; w3‡i;1†, …wi;3; w6;1; w3;i; wi‡3;1†, …w1;4; w6;1;

w4;1; w3;2†, …w2;4; w6;1; w5;1; w3;1†, …w1;4; w4;1; w2;4; w5;1† j 1  i  2g. Then B1[

B2[ B3 is a set of 4-cycles that packs K…4; 4; 2; 1; 1; 1† with leave L ˆ

ffw1;j; w2;jg; fwi;4; w3;ig j 1  j  3; 1  i  2g. &

Lemma 5.8. There exists a packing of K…4; 4; 1; 1; 1; 1; 1† with 4-cycles in which the leave L is six edges: two independent edges, and four more that induce a star. Proof. Let the partition of the vertex set be fi1j 1  i  4g, fi2j 1  i  4g, and

fzig for 1  i  5. Then on f31; 41; 32; 42; z1; . . . ; z5g we place a decomposition of K9

into 4-cycles, with one cycle being …31; 41; 42; 32†. The edges f31; 41g and f32; 42g;

are removed (since they do not belong to our original graph) and the edges f31; 32g,

f41; 42g become part of our leave L.

Now, on the bipartite graph K4;4 with parts f11; 21; 12; 22g and fz2; z3; z4; z5g, we

place a 4-cycle decomposition. We also take 4-cycles …11; 32; 21; 42†, …31; 12; 41; 22†;

and …11; 12; 21; 22†. The edges fx; z1g for x ˆ 11; 21; 12; 22 remain; these also form

Lemma 5.9. There exists a packing of K…4; 4; 1; 1; 1; 1; 1; 1; 1† with 4-cycles in which the leave is ®ve edges: three independent edges, and two more that induce a star.

Proof. Let the vertex set, with partition, be fi1 j 1  i  4g, fi2 j 1  i  4g, fzig

for 1  i  7. On the set f11; 12; zi j 1  i  7g we place a 4-cycle decomposition of

K9, ensuring that …11; z1; 12; z2† is one of the cycles; this cycle we remove. Call the set

of eight remaining cycles B1. We place the edges f11; z1g, f12; z1g into the leave L,

and retain edges f11; z2g, f12; z2g for later use. Next, on the bipartite graph K6;6with

vertex partition ff2i; 3i; 4ij i ˆ 1; 2g, fz1; z3; z4; z5; z6; z7gg, we place a 4-cycle

decomposition, say B2.

The remaining edges partition into ®ve further 4-cycles, B3, where

B3 ˆ f…z2; 11; 32; 21†; …z2; 31; 12; 41†; …z2; 12; 21; 42†; …z2; 22; 41; 32†; …11; 22; 31; 42†g;

leaving edges ffx1; x2g j 2  x  4g to form a further part of the leave. Now

B1[ B2[ B3 is a set of twenty two 4-cycles that pack K…4; 4; 1; 1; 1; 1; 1; 1; 1†, with

minimum leave L ˆ ff11; z1g; f12; z1g; fx1; x2g j 2  x  4g. &

Theorem 5.10. Suppose that G is a complete multipartite graph with s  1 even parts and t odd parts, where t is odd. Let the even sized parts have sizes v1  v2 


 vs, and let  ˆPsiˆ1vibe the number of vertices of odd degree in G.

There exists a maximum 4-cycle packing B of G with leave L in which either (A) jLj ₂‡ 3, or

(B) jLj  v1‡ 3.

Proof. As observed in the proof of Theorem 4.6, if L satis®es (A) or (B) then it is a minimum leave. Let E denote the set of vertices in the even parts and O the set of vertices in the odd parts.

First we deal with s ˆ 1, the case of only one even-sized part. We pack the odd parts with 4-cycles with leave as described in Remark 3.2, and then use condition () from E to O n fzg where z is one vertex in an odd-sized part. We possibly modify this set of 4-cycles, considering four cases in turn.

(i) t  1 (mod 8).

The current leave is a star that joins z to all vertices in E, so it has size v1;

hence L is minimum by (B). (ii) t  3 (mod 8).

The current leave is a star that joins z to all vertices in E, together with a K3

on three vertices from three different odd-sized parts, so it has size v1‡ 3.

Hence L is minimum by (B). (iii) t  5 (mod 8).

The current leave is a star centered at vertex z in O, together with (by choice) a K5 leave on a vertex set that includes vertex z and four other vertices, each

from a different odd-sized part. We apply Lemma 5.2 to a pair of vertices from the even-sized part together with the ®ve vertices of the K5 leave. The

together with a star that joins z to the remaining v1ÿ 2 vertices in E. This

leave has a total of v1‡ 2 edges, so it is a minimum leave, by (B).

(iv) t  7 (mod 8).

We proceed exactly as in the case t  5 (mod 8), but use Lemma 5.3 instead of Lemma 5.2. The ®nal leave contains v1‡ 1 edges and so is a minimum

leave by (B).

Next we assume that s  2, so there are at least two even-sized parts. Form a partition P of the vertices in E into pairs, the two vertices in each pair belonging to the same part. Partition as many of these pairs as possible into sets S1; . . . ; Syof size

four, so that each Sl; 1  l  y, contains at most two pairs from each part. Let S be

any of the remaining pairs. Then:

case (a) jSj  3, and we can assume that:

(ai) no two pairs in S occur in the same part, or (aii) Sl contains two pairs in V1, for 1  l  y.

case (b) jSj  4, and we can assume that:

(bi) all pairs in S except possibly one are in V1, and

(bii) Sl contains two pairs in V1, for 1  l  y.

To see that we can assume either (ai) or (aii) holds, suppose that W is a part containing two pairs p1 and p2 in S. If Sl contains two pairs in W for 1  l  y then

W ˆ V1 and (aii) holds. Otherwise one set, say S1, contains at most one pair in W.

Then since jS1j ˆ 4 and jSj  3, S1 contains a pair p3 contained in a part which

contains no pair in S. Interchange pairs p1and p3between S and S1. If W still contains

two pairs in S (so jSj ˆ 3), then this process can be repeated.

To see that (bi) and (bii) can be assumed we argue as follows. Clearly, the maximality of y forces all but at most one pair in S to occur together in one part, say Vi. If there exists a set Sl; 1  l  y, say S1, that contains at most one pair in Vi, then

replace one pair p1in S1that is not in Viwith a pair p2in S that is in Vito form S01. Let

S0_{ˆ …S [ fp}

1g† n fp2g. By the maximality of y, p1 is the only pair in S0that is not in

Vi (for otherwise we can form another set Sl‡1from the pairs in S0). Also, all sets S01

and Slfor 2  l  y contain exactly 2 pairs in Vi; for otherwise the above step could

be repeated, producing a second pair in S00_{not in V}

iwhich together with p1and two of

the jSj ÿ 2  2 pairs in S00 _{that are in V}

i form a set Sl‡1, again contradicting the

maximality of y. Therefore, (bii) holds, and so clearly Vi is the largest part, so i ˆ 1

and (bi) holds.

Let z 2 O. For each l, 1  l  y, let Slˆ fpl;ij 1  i  4g where possibly pl;2jÿ1

and pl;2j are subsets of the same part for j ˆ 1; 2. Let pl;i ˆ fw1_l;i; w2_l;ig. Let Bl be

formed from: the 4-cycles in a packing of K…4; 4; 1† (see Lemma 5.1) with partition fpl;1[ pl;2; pl;3[ pl;4; fzgg; and for j ˆ 1; 2, if pl;2jÿ1, and pl;2jare subsets of different

parts, include also the 4-cycle …w1

l;2jÿ1; w1l;2j; w2l;2jÿ1; w2l;2j†. Now the 4-cycles in Bl

cover all edges joining vertices inS4_iˆ1pl;i[ fzg that are in different parts, except for

the leave Ll that consists of a 1-factor joining vertices in pl;1[ pl;2 to vertices in

pl;3[ pl;4. Note that in case (b), by (bii) we have that each edge inSylˆ1Llis incident

with a vertex in V1.

Next, let B0

2be a set of 4-cycles that form a packing of K…vs‡1; . . . ; vs‡t† with leave

2 f1; 3; 5; 7g (see Remark 3.2). Then we pair off the vertices in O n fzg so that, in particular, fz2; z3g; . . . ; fz ÿ1; z g are pairs, and for each pair from O n fzg, together

with each pair in P (the partition of E into pairs), we take the induced 4-cycle, and place these 4-cycles in B0

1. Also, for each fp1; p2g  P such that

(1) p1 and p2 are in different parts, and

(2) for 1  l  y, fp1; p2g 6 Sl,

place the 4-cycle induced by p1[ p2in B03. (If t 6 1 (mod 8), note that not all of these

4-cycles will be in our ®nal maximum packing.)

Now consider the cases t  1; 3; 5; and 7 (mod 8) in turn. Let

B ˆ [y lˆ1 Bl ! [ [3 jˆ1 B0 j ! and L ˆ [y lˆ1 Ll ! [ E…R†;

where R is a star that joins z to each vertex in each pair of S. (i) t  1 (mod 8).

The cycles in B form a maximum packing with leave L which is a minimum leave because in case (a), jLj ˆ jEj=2 ‡ jSj  =2 ‡ 3 (see (A)), and in case (b), jLj ˆ v1

or v1‡ 2, according as all pairs in S are in V1, or one pair is not in V1 (see (B)).

For t  i (mod 8) when i ˆ 3; 5; or 7, we now start with B, that has leave L [ L0 i

where L0

iis a copy of Kide®ned on the vertex set fz ˆ z1; z2; . . . ; z g. We shall show

how to modify B to obtain a maximum packing. (ii) t  3 (mod 8).

If jSj ˆ 0 then jL0_{j ˆ}

2‡ 3, where L0ˆ L [ L03, and thus L0 is a minimum leave by

(A). Therefore we can assume that jSj  1.

If S contains two pairs p1 and p2 from different parts, then remove the 4-cycle

induced by p1[ p2 from B03, and remove the two 4-cycles joining p1[ p2 to fz2; z3g

from B0

1; then let B04 be the set of 4-cycles formed by applying Lemma 5.4 to

K7n fp1; p2g (regarding pi as an edge here), on the vertex set p1[ p2[

fzi j 1  i  3g. This results in the modi®ed leave L0 with

jL0_{j ˆ jLj ÿ 1 ˆ}



2‡ jSj ÿ 1 in case …a† …so 2  jSj  3† v1‡ 1 in case …b†:

8 < :

(See Figure 7.)

If all pairs in S belong to one part, say Vi (in case (b), i ˆ 1), then either there

exists a set, say S1, containing at most one pair in Vi(by (bii), this only arises in case

(a), and so by (ai) and (aii), jSj ˆ 1), or else each Sl contains two pairs in Vi, for

In the former case, if S1 contains a pair in Vi, let it be p1. If S1 contains one (or

two) pairs of pairs that are both subsets of the same part, then we can suppose that p3

and p4are in the same part (or p3and p4occur in one part, and p1and p2occur in one

part). Let p 2 S. Remove the 4-cycles in B1 from B. Remove from B01 the 4-cycles

induced by p [ fz2; z3g and by pi[ fz2; z3g for 1  i  4. Let B04 be the set of

4-cycles formed by applying Lemma 5.7 to K…4; 4; 2; 1; 1; 1† with vertex parts de®ned as follows. If p12 Vi then the parts are p [ p1; p3[ p4; p2; fz1g; fz2g; fz3g, in

which case if p3 and p4 are in different parts we add to B04 the 4-cycle formed by the

edges joining p3 to p4. If p1 62 Vi then the parts are p1[ p2; p3[ p4; p;

fz1g; fz2g; fz3g, in which case for i ˆ 1; 2, if p2iÿ1 and p2i are in different parts

then we also add to B0

4 the 4-cycle formed by the edges joining p2iÿ1 to p2i. The

modi®ed leave L0 _satis®es

jL0_{j ˆ}

2‡ jSj ÿ 1 ˆ 

2; …1†

so L0 _{is a minimum leave by (A). (See Figure 8.)}

In the latter case, every edge in L meets V1, so jL [ L03j ˆ v1‡ 3 and thus

L0_{ˆ L [ L}0

3 is already a minimum leave by (B).

(iii) t  5 (mod 8).

If jSj ˆ 0 then clearly y  1, so let S1 ˆ fp1; p2; p3; p4g, where we can assume for

i ˆ 1; 2 and j ˆ 3; 4 that pi and pjoccur in different parts. Replace the 4-cycles in B01 FIG. 7.

joining pi, 1  i  4, to fzj j 2  j  5g and the 4-cycles in B1 with the set B04 of

4-cycles de®ned as follows. Let B0

4 contain the 4-cycles formed by applying Lemma

5.8 to K…4; 4; 1; 1; 1; 1; 1† with parts p1[ p2, p3[ p4 and fzig for 1  i  5, together

with the 4-cycles that join p2iÿ1to p2i, i ˆ 1; 2, whenever p2iÿ1and p2iare in different

even-sized parts. The resulting leave L0 _{satis®es jL}0_{j ˆ}

2‡ 2, so is a minimum leave

by (A). Therefore, we can assume that jSj  1.

If S contains two pairs p1 and p2 from different parts, then remove the 4-cycle

induced by p1[ p2 from B03 and the four 4-cycles joining p1[ p2 to fz2; z3; z4; z5g

from B0

1. Let B04be the set of 4-cycles formed by applying Lemma 5.5 to K9n fp1; p2g

(regarding pias an edge here) on the vertex set p1[ p2[ fzij 1  i  5g. This results

in the modi®ed leave

jL0_{j ˆ jLj ÿ 2 ˆ}



2‡ jSj ÿ 2 in case …a† …so jSj ˆ 2 or 3†; v1 in case …b†:

8 < : (See Figure 9.)

Now if all pairs in S belong to one part, say Vi, then let p 2 S and remove the two

4-cycles joining p to fz2; z3; z4; z5g from B01. Then apply Lemma 5.2 to K7ÿ fpg with

vertex set p [ fz1; z2; . . . ; z5g. If jSj ˆ 1, then jL0j ˆ₂‡ 3, so L0 is minimum by (A).

If jSj  2, and all pairs in S belong to Vi, then since (ai) is not satis®ed, each Sl

contains two pairs in Vifor 1  l  y (so i ˆ 1). Therefore, every edge in L meets V1,

so we have ®nal leave L0 _{satisfying jL}0_{j ˆ v} 1‡ 2.

(iv) t  7 (mod 8).

If jSj ˆ 0 then proceed exactly as in the case where t  5 (mod 8) and jSj ˆ 0, except that Lemma 5.9 is used instead of Lemma 5.8. Then the resulting modi®ed leave L0

satis®es jL0_{j ˆ}

2‡ 1, so is a minimum leave by (A). Therefore, we can assume that

jSj  1.

If jSj ˆ 2 or jSj  4 and S contains two pairs p1 and p2 from two different parts,

then remove: the 4-cycle induced by p1[ p2 from B03, and the six 4-cycles joining

p1[ p2 to fzi j 2  i  7g from B01. Then apply Lemma 5.6 to K11ÿ fp1; p2g on the

vertex set p1[ p2[ fzi j 1  i  7g to obtain a set of 4-cycles B04. This results in the

modi®ed leave L0 _with:

jL0_{j ˆ}  2‡ 3 if jSj ˆ 2; and v1‡ 3 if jSj  4: 8 < : (See Figure 10.) FIG. 9.

If jSj 6ˆ 3, and all pairs occur in one part (so if jSj  2, then in fact jSj ˆ 1 by (ai)), apply Lemma 5.3 to K9ÿ p with vertex set fzij 1  i  7g [ p, where p 2 S. The

resulting leave L0 _satis®es

jL0_{j ˆ}  2‡ jSj ‡ 1 ˆ  2‡ 2; if jSj  2; v1‡ 1; if jSj  4: 8 < :

So L0 _{is a minimum leave. (See Figure 11.)}

Finally, suppose jSj ˆ 3. Let S ˆ fp1; p2; p3g.

(1) If all three of these pairs in S belong to different parts, then remove the 4-cycles in B0

3 joining pito pjfor 1  i < j  3, and remove from B01 the nine

4-cycles that join p1[ p2[ p3 to fzij 2  i  7g. Let B04 be a packing of K13

on the vertex set fp1; p2; p3; zij 1  i  7g, with leave being the 6-cycle

…p1

1; p12; p22; p13; p32; p21†, where pair pi ˆ fp1i; p2ig. This is equivalent to a 4-cycle

packing of K…2; 2; 2; 1; 1; 1; 1; 1; 1; 1† with three disjoint edges in the leave, between vertices in the parts of size 2. Thus the ®nal leave L0 _satis®es

jL0_{j ˆ =2:}

(2) If exactly two pairs, p1 and p3, belong to one part, W1 and p2 to a third part,

then (ai) is not satis®ed. Therefore, each Sl, 1  l  y, contains two pairs from

W1, so W1 ˆ V1. Remove the 4-cycle in B03joining p1 to p2, and remove from

B0

1 the six 4-cycles that join p1[ p2 to fzij 2  i  6g. Let B04 be the set of FIG. 10.

4-cycles formed by applying Lemma 5.6 to K11n fp1; p2g, on the vertex set

fzi j 1  i  7g [ p1[ p2. This packing of K11n fp1; p2g has the leave of two

disjoint edges between p1and p2 and a K3 on the vertex set fz1; z2; z3g, so the

®nal leave L0 _{satis®es jL}0_{j ˆ v} 1‡ 3.

(3) Finally, we can assume that jSj ˆ 3 and all three pairs in S occur in the same part, say W. Since (ai) is not satis®ed, each Sl contains two pairs from W,

for 1  l  y, so W ˆ V1. Then using Lemma 5.3 on K9n fp1g yields

jL0_{j ˆ v}

1‡ 1. &

6. CONCLUDING REMARKS

We now have the following result.

Theorem 6.1. Let G be a complete multipartite graph with  vertices of odd degree and  vertices in the largest part containing vertices of odd degree (if such a part exists). Then there exists a maximum 4-cycle packing of G with leave L satisfying

(i) maxf=2; g  jLj  maxf=2; g ‡ 3, if G does not have n parts all of odd size with n  5 or 7 (mod 8); and

(ii) jLj ˆ 6 or 5 if G has n parts, all of odd size, with n  5 or 7 (mod 8), respectively.

Remark: Note that the size of the leave is completely determined by the inequalities.

Proof. Clearly, jLj  =2 and jLj  . Also, for any other 4-cycle packing of G with leave L0_{, 4 must divide jL}0_{j ÿ jLj. Therefore, the result follows from the 4-cycle}

packing of G with leave L that is constructed in one of Lemmas 2.2 and 3.1 and Theorems 4.6 and 5.10. &

We remark that of course many different minimum leaves are possible in most cases. For instance, in cases where a component of the leave as described above is a star with center at a vertex in an odd part, this could be split into several smaller stars having centers at different vertices in odd sized parts.

The related problem of packing a -fold complete multipartite graph will be the subject of a subsequent paper, for reasons of length.

ACKNOWLEDGMENT

This research was supported by an Australian Research Council grant to E.J.B (grant number A69701550), NSC grant to H-L.F (grant number 88-2115-M-009-013, and to C.A.R a NSF grant (DMS-9531722) and an ONR grant (N00014-97-1-1067).

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