The metamorphosis of λ -fold K4-e designs into maximum packings of λKn with 4-cycles, λ⩾2

Journal of Statistical Planning and Inference 138 (2008) 3316 -- 3325

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The metamorphosis of -fold K4 − e designs into maximum packings of Kn with 4-cycles,  2夡 C.C. Lindnera,∗ , Antoinette Tripodib a Department b Department

of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA of Mathematics, University of Messina, Contrada Papardo, 31-98166 Sant'Agata, Messina, Italy

ARTICLE

INFO

Article history: Received 13 January 2005 Accepted 20 July 2005 Available online 18 March 2008 Keywords: K4 − e design 4-cycle system Maximum packing Metamorphosis

ABSTRACT

Let K4 − e = . If we remove the "diagonal'' edge the result is a 4-cycle. Let (X, B) be a -fold K4 − e design of order n; i.e., a decomposition of Kn into copies of K4 − e. Let D(B) be the collection of "diagonals'' removed from the graphs in B and C1 (B) the resulting collection of 4-cycles. If C2 (B) is a reassembly of these edges into 4-cycles and L is the collection of edges in D(B) not used in a 4-cycle of C2 (B), then (X, C1 (B)∪C2 (B), L) is a packing of Kn with 4-cycles and is called a metamorphosis of (X, B). In Lindner and Tripodi [2005. The metamorphosis of K4 − e designs into maximum packings of Kn with 4-cycles. Ars Combin. 75, 333--349.] a complete solution is given for the existence problem of K4 − e designs ( = 1) having a metamorphosis into a maximum packing of Kn with all possible leaves. The purpose of this paper is the complete solution of the above problem for all values of  > 1. © 2008 Elsevier B.V. All rights reserved.

1. Introduction A -fold K4 − e design of order n is a pair (X, B), where B is a collection of copies of the graph

which partitions the edge set of Kn ( copies of the complete undirected graph on n vertices) with vertex set X. When  =1 we will simply say K4 −e design. It is well known that the spectrum for -fold K4 −e designs is precisely the set of all (i) n ≡ 0, 1 (mod 5)  6 for  = 1, (ii) n ≡ 0, 1 (mod 5) for  ≡ 1, 2, 3, or 4 (mod 5)  2, and (iii) n  4 for  ≡ 0 (mod 5). A -fold 4-cycle system of order n is a pair (X, C), where C is a collection of 4-cycles which partitions the edge set of Kn with vertex set X. When  = 1 we will simply say 4-cycle system. It is well known that the spectrum for -fold 4-cycle systems is precisely the set of all (i) n ≡ 1 (mod 8) for  ≡ 1, 3 (mod 4), (ii) n ≡ 0, 1 (mod 4) for  ≡ 2 (mod 4), and (iii) n  4 for  ≡ 0 (mod 4). In both of the above definitions if we drop the quantification "partitions'' we have the definition of a -fold partial K4 − e design and a -fold partial 4-cycle system.

夡 ∗

Supported in part by M.U.R.S.T. "Strutture geometriche, combinatoria e loro applicazioni'', COFIN. 2003, and I.N.D.A.M. (G.N.S.A.G.A.). Corresponding author. Tel.: +1 334 844 3747. E-mail addresses: [email protected] (C.C. Lindner), [email protected] (A. Tripodi).

0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2005.07.013

C.C. Lindner, A. Tripodi / Journal of Statistical Planning and Inference 138 (2008) 3316 -- 3325

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In what follows we will denote the m-cycle with edges {x1 , x2 }, {x2 , x3 }, . . . , {xm−1 , xm }, {xm , x1 } by any cyclic shift of (x1 , x2 , x3 , . . . , xm ) or (x2 , x1 , xm , xm−1 , . . . , x3 ); and the graph d

b

a

c

K4 − e =

by any one of [a, b, c, d], [a, b, d, c], [b, a, c, d], or [b, a, d, c]. A packing of Kn with 4-cycles is a triple (X, C, L), where (X, C) is a partial -fold 4-cycle system and L is the set of edges not belonging to a 4-cycle in C. The collection of edges L is called the leave of the packing. If |C| is as large as possible or, equivalently, |L| is as small as possible, the packing (X, C, L) is said to be maximum. (So, for example, a -fold 4-cycle system of order n is a maximum packing of Kn with leave the empty set.) Now let (X, B) be a -fold K4 − e design of order n and let D(B) = {{a, b} | [a, b, c, d] ∈ B} and C1 (B) = {(a, c, b, d) | [a, b, c, d] ∈ B}. Then (X, C1 (B)) is a -fold partial 4-cycle system. If the edges belonging to D(B) can be arranged into a collection of 4-cycles C2 (B) with leave L, then (X, C1 (B) ∪ C2 (B), L) is a packing of Kn with 4-cycles, and is said to be a metamorphosis of (X, B). (The algorithm of going from (X, B) to (X, C1 (B) ∪ C2 (B), L) is also called a metamorphosis.) In Lindner and Tripodi (2005) the spectrum for K4 − e designs having metamorphoses into maximum packings of Kn with 4-cycles with all possible leaves is completely determined. The purpose of this paper is the complete solution of the problem of constructing for each  > 1 and for each admissible value of n a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves.

2. Preliminaries The necessary conditions for a leave L of a maximum packing of Kn with 4-cycles are: if (n − 1) ≡ 0 (mod 2), L must be a graph  of  even degree; if (n − 1) ≡ 1 (mod 2), L is a spanning graph of odd degree; and the number of edges of L must be congruent to  2n (mod 4). It is well known that a maximum packing of Kn with 4-cycles has leave: (i) a 1-factor if n is even; (ii) the empty set if n ≡ 1 (mod 8); (iii) a 3-cycle if n ≡ 3 (mod 8); (iv) a graph of even degree with 6 edges (i.e., two disjoint 3-cycles, two 3-cycles with a common vertex (a bowtie), or a 6-cycle) if n ≡ 5 (mod 8); and (v) a 5-cycle if n ≡ 7 (mod 8) (see Fig. 1). Table 1 shows the possible leaves of a maximum packing of Kn with 4-cycles for  > 1. In Lindner and Tripodi (2005) the following theorem is proved. Theorem 2.1. There exists a K4 − e design of every order n ≡ 0, 1 (mod 5)  6, n  = 11, having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves. (There exists a maximum packing of K11 with 4-cycles, but it cannot be obtained from a K4 − e design.)

1 (mod 8)

0, 2, 4, or 6 (mod 8)

3 (mod 8)

5 (mod 8)

∅ .. .

Fig. 1. Possible leaves of a maximum packing of Kn with 4-cycles.

7 (mod 8)

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Table 1 Possible leaves of a maximum packing of Kn with 4-cycles for  > 1

 (mod 4) > 1

n (mod 8) 0 1 2 3 4 5 6 7 G

0

1

2

3

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅

1-Factor ∅ 1-Factor Triangle 1-Factor Double edge 1-Factor H

∅ ∅ Double edge Double edge ∅ ∅ Double edge Double edge

1-Factor ∅ G H 1-Factor Double edge G Triangle

is a graph on n vertices with (n + 4)/2 edges and odd vertex degrees (see Fig. 2); H is a graph with 5 edges and even vertex degrees (see Fig. 3).

In the subsequent sections we will give a complete solution of the problem of constructing for each  > 1 and for each admissible value of n a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves. It will be sufficient to give solutions for  = 2, 3, 4, and  ≡ 0 (mod 5), since these results can be pasted together to obtain a complete solution for all other values of . The results will be organized into two sections followed by a summary. Since the case  ≡ 0 (mod 5) is by far the more difficult to handle, a whole section is dedicated to it. 3. The cases  = 2, 3, 4 In this section we will show that for every  = 2, 3, 4 and for every n ≡ 0, 1 (mod 5) there exists a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves (for the leaves see Table 1). 3.1. The case  = 2 To begin with, we will give examples for n = 5 and 11. Example 3.1 (n=5). Let (Z5 , B) be the twofold K4 −e design where B={[1, 2, 4, 0], [2, 3, 1, 0], [3, 4, 2, 0], [1, 4, 3, 0]}. Then (X, C1 (B)∪ C2 (B)) is a twofold 4-cycle system, where C2 (B) = {(1, 2, 3, 4)}. Example 3.2 (n = 11). Let F = {F0 , F1 , F2 , F3 , F4 } be a 1-factorization of K6 with vertex set X1 . Set X = X1 ∪ Z5 and define a collection B of copies of K4 − e as follows: (1) Let (X1 , B1 ) be a K4 − e design of order 6 having a metamorphosis into a maximum packing of K6 with leave L1 = {{x1 , x2 }, {x3 , x4 }, {x5 , x6 }} (see Lindner and Tripodi, 2005); put B1 ⊆ B. (2) Let (Z5 , B2 ) be the twofold K4 − e design of order 5 in Example 3.1; put B2 ⊆ B. (3) [a, b, i, i + 1] ∈ B for all {a, b} ∈ Fi , i = 0, 1, 2, 3, 4. Then (X, B) is a twofold K4 − e design of order 11. The metamorphosis is the following: use the metamorphosis in (1) and (2); delete the edges {a, b}, [a, b, i, i + 1] ∈ B, from the type (3) graphs and, since this gives a copy of K6 , replace these edges with a maximum packing of K6 with 4-cycles with leave L2 = {{x2 , x3 }, {x1 , x4 }, {x5 , x6 }}. Combining the leaves L1 and L2 gives the 4-cycle (x1 , x2 , x3 , x4 ) and the double edge {{x5 , x6 }, {x5 , x6 }}. Lemma 3.1. For every n ≡ 0, 1 (mod 5) there exists a twofold K4 −e design of order n having a metamorphosis into a maximum packing of 2Kn with 4-cycles. Proof. Examples 3.1 and 3.2 take care of the cases n = 5 and 11, so we can assume n  6, n  = 11. Take two K4 − e designs of order n, (X, B1 ) and (X, B2 ), with X = {1, 2, . . . , n}, having metamorphoses into maximum packings of Kn with 4-cycles. Then (X, B1 ∪ B2 ) is a twofold K4 − e design of order n. The metamorphosis is the following: use metamorphoses of (X, B1 ) and (X, B2 ) with suitable leaves L1 and L2 , respectively, and combine L1 and L2 as the case may be. (a) n ≡ 0, 4 (mod 8). Let L1 = {{1, 2}, {3, 4}, {5, 6}, {7, 8}, . . . , {n−3, n−2}, {n−1, n}} and L2 ={{1, 4}, {2, 3}, {5, 8}, {6, 7}, . . . , {n−3, n}, {n − 2, n − 1}}. Reassemble the edges of L1 and L2 into the n/4 4-cycles (1, 2, 3, 4), (5, 6, 7, 8), . . . , (n − 3, n − 2, n − 1, n). (b) n ≡ 1 (mod 8). L1 = L2 = ∅ and so the result is a maximum packing of 2Kn with 4-cycles with leave the empty set.

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(c) n ≡ 2, 6 (mod 8). Let L1 ={{1, 2}, {3, 4}, {5, 6}, {7, 8}, . . . , {n−5, n−4}, {n−3, n−2}, {n−1, n}} and L2 ={{1, 4}, {2, 3}, {5, 8}, {6, 7}, . . . , {n−5, n−2}, {n−4, n−3}, {n−1, n}}. Reassemble the edges of L1 and L2 into the (n−2)/4 4-cycles (1, 2, 3, 4), (5, 6, 7, 8), . . . , (n − 5, n − 4, n − 3, n − 2) and the double edge {{n − 1, n}, {n − 1, n}}. (d) n ≡ 3 (mod 8). Let L1 = {(1, 2, 3)} and L2 = {(1, 2, 4)}. Combining L1 and L2 gives the 4-cycle (1, 3, 2, 4) and the double edge {{1, 2}, {1, 2}}. (e) n ≡ 5 (mod 8). Let L1 ={(1, 2, 3, 4, 5, 6)} and L2 ={(3, 7, 8, 4, 9, 6)}. Combining L1 and L2 gives the 4-cycles (1, 2, 3, 6), (3, 7, 8, 4), (4, 5, 6, 9). (f) n ≡ 7 (mod 8). Let L1 = {(1, 2, 3, 4, 5)} and L2 = {(1, 6, 7, 2, 5)}. Combining L1 and L2 gives the 4-cycles (2, 3, 4, 5), (1, 2, 7, 6) and the double edge {{1, 5}, {1, 5}}.  3.2. The case  = 3 To begin with, we will give examples for n = 5 and 11. Example 3.3 (n = 5). Let (X, B) be the threefold K4 − e design where B = {[1, 2, 3, 5], [2, 3, 4, 5], [3, 4, 2, 5], [1, 4, 3, 2], [1, 5, 3, 4], [1, 5, 2, 4]}. Then (X, C1 (B) ∪ C2 (B), L) is a threefold 4-cycle system, where C2 (B) = {(1, 2, 3, 4)} and L = {{1, 5}, {1, 5}}. Example 3.4 (n = 11, with all possible leaves). Let X = X1 ∪ {∞}, where X1 = {1, 2, . . . , 10}. Define a collection B of copies of K4 − e as follows: (1) Let (X1 , B1 ) be a twofold K4 − e design of order 10 having a metamorphosis into a packing of 2K10 with 4-cycles; put B1 ⊆ B. (2) Place the 15 copies of K4 − e[∞, 6, 2, 9], [∞, 2, 4, 7], [∞, 1, 2, 10], [∞, 5, 8, 9], [∞, 8, 1, 7], [∞, 7, 5, 10], [∞, 3, 8, 9], [∞, 4, 5, 10], [4, 6, 8, ∞], [1, 3, 4, ∞], [6, 3, 7, ∞], [2, 3, 5, 10], [5, 6, 1, 10], [8, 9, 2, 10], [7, 9, 1, 4] in B. Then (X, B) is a threefold K4 − e design of order 11. The metamorphosis is the following: use the metamorphosis in (1) with leave the double edge L1 ; delete the diagonals from the type (2) graphs and rearrange them into the 4-cycles (∞, 1, 3, 2), (∞, 4, 6, 5), (∞, 7, 9, 8), and the triangle L2 = (∞, 3, 6). (X, C1 (B) ∪ C2 (B), L1 ∪ L2 ) is a maximum packing of 3K11 , where L1 ∪ L2 is isomorphic to Hi , i = 1, 2, or 3 (see Fig. 3), if |V(L1 ) ∩ V(L2 )| = 0, 1, or 2, respectively. There are four possible leaves for a threefold 4-cycle system of order 11; the fourth is a 5-cycle. The following is a solution for a 5-cycle. Replace B1 in (1) with B1 ∪ B1 , where B1 and B1 are the collections of graphs of two K4 − e designs of order 10 having metamorphoses into maximum packings of K10 with 4-cycles with leaves L1 = {{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}} and L1 = {{2, 3}, {4, 6}, {1, 5}, {7, 10}, {8, 9}}, respectively. Combining L1 , L1 , and L2 gives the 4-cycles (∞, 3, 4, 6) and (7, 8, 9, 10) and the 5-cycle (1, 2, 3, 6, 5). Lemma 3.2. For every n ≡ 0, 1 (mod 5) there exists a three fold K4 − e design of order n having a metamorphosis into a maximum packing of 3Kn with 4-cycles with all possible leaves. Proof. Examples 3.3 and 3.4 take care of the cases n = 5 and 11, so we can assume n  6, n  = 11. Let X = {1, 2, . . . , n} be the vertex set. (a) n ≡ 0, 1, 4 (mod 8). Paste together a solution for  = 1 and a solution for  = 2. (b) n ≡ 2, 6 (mod 8). To obtain a solution with leave G1 or G2 (see Fig. 2) paste together a solution for  = 1 with leave {{1, 2}, {3, 4}, . . . , {n − 1, n}} and a solution for  = 2 with leave {{1, 2}, {1, 2}} or {{2, 3}, {2, 3}}. For the remaining leaves paste together three solutions for  = 1, (X, B1 ), (X, B2 ), and (X, B3 ), with leaves the 1-factors L1 , L2 , and L3 , respectively, where L2 ⊇ A={{11, 12}, {13, 14}, {15, 16}, {17, 18}, . . . , {n−3, n−2}, {n−1, n}}, and L3 ⊇ A ={{12, 13}, {11, 14}, {16, 17}, {15, 18}, . . . , {n− 1, n−2}, {n−3, n}}. Combine A and A to obtain the (n−10)/4 4-cycles (11, 12, 13, 14), (15, 16, 17, 18), . . . , (n−3, n−2, n−1, n). Rearrange the remaining edges of L1 , L2 , and L3 as the case may be. (1) Let L1 ⊇ {{1, 2}, {5, 6}, {3, 4}, {7, 8}}, L2 \A={{1, 3}, {5, 7}, {2, 6}, {4, 8}, {9, 10}}, and L3 \A ={{1, 4}, {5, 8}, {3, 7}, {2, 9}, {6, 10}}. Combining L1 , L2 \A, and L3 \A gives the 4-cycles (3, 4, 8, 7) and (2, 6, 10, 9), the two 3-stars {{1, 2}, {1, 3}, {1, 4}} and {{5, 6}, {5, 7}, {5, 8}}, and the (n − 8)/2 disjoint edges of L1 \{{1, 2}, {5, 6}, {3, 4}, {7, 8}}. (2) Let L1 ⊇ {{1, 2}, {4, 5}, {3, 6}}, L2 \A = {{1, 3}, {4, 6}, {2, 5}, {7, 8}, {9, 10}}, and L3 \A = {{1, 4}, {2, 3}, {5, 6}, {8, 9}, {7, 10}}. Combining L1 , L2 \A, and L3 \A gives the 4-cycles (2, 3, 6, 5) and (7, 8, 9, 10), the graph {{1, 2}, {1, 3}, {1, 4}, {4, 5}, {4, 6}}, and the (n − 6)/2 disjoint edges of L1 \{{1, 2}, {4, 5}, {3, 6}}. (3) Let (1, 2, 3, 4) ∈ C2 (B1 ), L2 \A = {{1, 5}, {4, 7}, {2, 6}, {3, 8}, {9, 10}}, and L3 \A = {{1, 6}, {5, 4}, {2, 7}, {3, 9}, {8, 10}}. There are three possibilities: the vertices 1, 2, 3, and 4 belong to four different pairs of L1 ; two vertices of (1, 2, 3, 4), suppose 1 and 3, form a pair of L1 and the remaining two vertices belong to different pairs of L1 ; or {1, 3}, {2, 4} ∈ L1 . So we can assume L1 ⊇ {{1, 7}, {5, 2}, {4, 6}}; L1 ⊇ {{1, 3}, {5, 2}, {4, 6}}, or L1 ⊇ {{1, 3}, {4, 2}, {5, 6}}, respectively. Combining (1, 2, 3, 4), L1 , L2 \A, and L3 \A gives: the 4-cycles (7, 2, 3, 4), (2, 6, 4, 5), and (3, 8, 10, 9), the 5-star {{1, 2}, {1, 7}, {1, 4}, {1, 5}, {1, 6}},

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G1

...

G2

...

...

G3

...

G4

...

G5

Fig. 2. All possible graphs on n vertices with (n + 4)/2 edges and odd vertex degrees.

C5

H1

H2

H3

Fig. 3. All possible graphs with 5 edges and even vertex degrees.

and the (n − 6)/2 disjoint edges of L1 \{{1, 7}, {5, 2}, {4, 6}}; or the 4-cycles (7, 2, 3, 4), (2, 6, 4, 5) and (3, 8, 10, 9), the 5star {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}}, and the (n−6)/2 disjoint edges of L1 \{{1, 3}, {5, 2}, {4, 6}}; or the 4-cycles (7, 2, 3, 4), (2, 4, 5, 6) and (3, 8, 10, 9), the 5-star {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}}, and the (n − 6)/2 disjoint edges of L1 \{{1, 3}, {4, 2}, {5, 6}}, respectively. (c) n ≡ 3 (mod 8). Paste together a solution for  = 1 with leave the triangle L1 and a solution for  = 2 with leave the double edge L2 . The result is a solution for  = 3 with leave L = L1 ∪ L2 which is isomorphic to Hi , i = 1, 2, or 3 (see Fig. 3), if |V(L1 ) ∩ V(L2 )| = 0, 1, or 2, respectively. To obtain a solution for a 5-cycle paste together three solutions for  = 1 with leaves L1 = {(1, 2, 3)}, L2 = {(2, 4, 5)}, and L3 = {(3, 4, 6)}, respectively. Combining L1 , L2 , and L3 gives the 4-cycle (2, 3, 6, 4) and the 5-cycle (1, 3, 4, 5, 2). (d) n ≡ 5 (mod 8). Paste together three solutions for  = 1 with leaves L1 = {(1, 2, 3), (1, 4, 5)}, L2 = {(6, 3, 5), (6, 7, 8)}, and L3 ={(7, 8, 9), (1, 5, 10)}, respectively. Combining L1 , L2 , and L3 , gives the 4-cycles (10, 1, 4, 5), (1, 2, 3, 5), (1, 3, 6, 5), (6, 7, 9, 8), and the double edge {{7, 8}, {7, 8}}. (e) n ≡ 7 (mod 8). Paste together a solution for  = 1 with leave L1 = {(1, 2, 3, 4, 5)} and a solution for  = 2 with leave L2 = {{1, 3}, {1, 3}}. Combining L1 and L2 gives the 4-cycle (1, 3, 4, 5) and the triangle (1, 2, 3).  3.3. The case  = 4 To begin with, we will give an example for n = 11.

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Example 3.5 (n = 11). Paste together four K4 − e designs with vertex set Z11 having metamorphoses into packings of K11 with leaves the 11-cycles Li = (0, 1, 2, . . . , 10), for i = 1, 2, 3, and L4 = (0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8), respectively, (for examples see Gionfriddo et al., 1991). Combining Li , i = 1, 2, 3, 4, gives the collection of 4-cycles {(i, 1 + i, 2 + i, 3 + i) | i ∈ Z11 }. Lemma 3.3. For every n ≡ 0, 1 (mod 5) there exists a fourfold K4 − e design of order n having a metamorphosis into a maximum packing of 4Kn with 4-cycles. Proof. Example 3.4 takes care of the case n = 11. Let n  5, n  = 11, and X = {1, 2, . . . , n} be the vertex set. (a) n ≡ 0, 1, 4, 5 (mod 8). Paste together two solutions for  = 2. (b) n ≡ 2, 6 (mod 8). Paste together two solutions for =1 with leaves the 1-factors L1 ={{2, 3}, {1, 4}, {5, 6}, {7, 8}, {9, 10}, {11, 12}, {13, 14}, . . . , {n − 3, n − 2}, {n − 1, n}} and L2 = {{3, 4}, {2, 5}, {1, 6}, {8, 9}, {7, 10}, {13, 12}, {11, 14}, . . . , {n − 1, n − 2}, {n − 3, n}}, respectively, and one solution for  = 2 with leave L3 = {{1, 2}, {1, 2}}. Combining L1 , L2 , and L3 gives the (n − 6)/4 4-cycles (1, 2, 3, 4), (1, 2, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14), . . . , (n − 3, n − 2, n − 1, n). (c) n ≡ 3 (mod 8). Paste together a solution for  = 1 with leave {(1, 2, 3)} and a solution for  = 3 with leave {(1, 4, 2, 5, 6)}. Rearrange the leaves into the 4-cycles (1, 2, 5, 6) and (1, 4, 2, 3). (d) n ≡ 7 (mod 8). Paste together a solution for  = 1 with leave {(1, 4, 2, 5, 6)} and a solution for  = 3 with leave {(1, 2, 3)} and combine the leaves as in (c).  4. The case  ≡ 0 (mod 5) When  ≡ 0 (mod 5), the spectrum for -fold K4 − e designs is precisely the set of all n  4; i.e., no restrictions. The following Folk construction packs the spectrum (see Gionfriddo et al., 1991). Folk construction: Let (P, ◦) be an idempotent anti-symmetric (a ◦ b  = b ◦ a, a  = b ∈ P) quasigroup of order n  4. Let B = {[a, b, a ◦ b, b ◦ a]| all a  = b ∈ P}. Then (P, B) is a fivefold K4 − e design of order n. Taking k copies of (P, B) produces a 5k-fold K4 − e design of order n. In this section we will show that for every  ≡ 0 (mod 5) and for every n  4 there exists a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves (for the leaves see Table 1). We begin with the case  = 5. Lemma 4.1. For every n ≡ 0, 1 (mod 5) there exists a fivefold K4 −e design of order n having a metamorphosis into a maximum packing of 5Kn with 4-cycles with all possible leaves. Proof. Assume n ≡ 0, 1 (mod 5). Paste together: (a) a solution for  = 1 and a solution for  = 4, when n ≡ 0, 1, 2, 3, 4, 6, 7 (mod 8) (note that if n ≡ 7 (mod 8) this gives a solution for  = 5 with leave a 5-cycle); (b) a solution for  = 2 and a solution for  = 3, when n ≡ 5, 7 (mod 8) (note that if n ≡ 7 (mod 8) this gives a solution for  = 5 with leave Hi , i = 1, 2, or 3).  The following lemma takes care of the cases n ≡ 0, 1, 2, 3, 4, 6 (mod 8). Lemma 4.2. For every n ≡ 0, 1, 2, 3, 4, 6 (mod 8) there exists a fivefold K4 − e design of order n having a metamorphosis into a maximum packing of 5Kn with 4-cycles. Proof. Let (P, B) be a fivefold K4 − e design of order n ≡ 0, 1, 2, 3, 4, 6 (mod 8) obtained by the Folk construction. Then D(B) is a copy of Kn . If we replace D(B) with a maximum packing of Kn with 4-cycles (P, C, L), then (P, C1 (B) ∪ C, L) is a maximum packing of 5Kn .  Remark 4.1. When n ≡ 7 (mod 8), by arguments similar to the proof of Lemma 4.2 the Folk construction yields a fivefold K4 − e design having a metamorphosis into a maximum packing of 5Kn with 4-cycles with leave a 5-cycle. However, for a maximum packing of 5Kn with 4-cycles there are three other possible leaves (see Table 1). Lemmas 4.1 and 4.2 and Remark 4.1 leave the cases n ≡ 7, 23, 39 (mod 40) to complete with solutions for the leaves Hi , i=1, 2, 3, and n ≡ 13, 29, 37 (mod 40) with leave a double edge. We will begin with some examples for small orders followed by six recursive constructions for the remaining cases.

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Example 4.1 (n = 7 with leave Hi , i ∈ {1, 2, 3}). Let X = X1 ∪ {∞}, where X1 = {1, 2, 3, 4, 5, 6}. Define a collection B of copies of K4 − e as follows: (1) Let (X1 , B1 ) be a twofold K4 − e design of order 6 having a metamorphosis into a maximum packing of 2K6 with 4-cycles with leave the double edge L1 (see Section 3); put B1 ⊆ B. − e [∞, 5, 6, 2], [∞, 6, 5, 4], [4, 5, 1, ∞], [4, 6, 2, ∞], (2) Place the 15 copies of K4 [1, 3, 5, ∞], [3, 5, 6, ∞], [4, 5, 2, ∞], [1, 4, 5, ∞], [1, 2, 6, ∞], [1, 3, 4, 2], [2, 3, 5, ∞], [3, 6, 4, ∞], [1, 6, 3, ∞], [2, 3, 4, ∞], [1, 2, 6, ∞] in B. Then (X, B) is a fivefold K4 − e design of order 7. The metamorphosis is the following: use the metamorphosis in (1); delete the diagonals from the type (2) graphs and rearrange them into the 4-cycles (1, 3, 5, 4), (∞, 5, 4, 6), (1, 2, 3, 6), and the triangle L2 = (1, 2, 3). Then (X, C1 (B) ∪ C2 (B), L1 ∪ L2 ) is a maximum packing of 5K7 , where L1 ∪ L2 is isomorphic to Hi , i = 1, 2, or 3 (see Fig. 3), if |V(L1 ) ∩ V(L2 )| = 0, 1, or 2, respectively. In order to handle the case n = 13 we need the following example. Example 4.2 (A fivefold K4 − e design with holes of size 2 having a metamorphosis into a packing of 5K2,2,2 with 4-cycles with leave the empty set). Let K2,2,2 have parts {x0 , x1 }, {y0 , y1 }, and {z0 , z1 } and let B = {[a, b, a ◦ b, b ◦ a]} where a ◦ b and b ◦ a are computed in the following quasigroup with holes {x0 , x1 }, {y0 , y1 }, and {z0 , z1 }.

Then C1 (B) ∪ C2 (B), where C2 (B) = {(x0 , y0 , x1 , y1 ), (x0 , z0 , x1 , z1 ), (z0 , y0 , z1 , y1 )}, partitions the edge set of 5K2,2,2 (with parts {x0 , x1 }, {y0 , y1 }, {z0 , z1 }) into 4-cycles. Finally, before giving a solution for n = 13 we need some preliminaries. Two collections of graphs P1 and P2 are said to be balanced provided they contain exactly the same edges. The following collections of graphs are balanced: Ai = {((2i − 1)0 , (2i + 1)0 , (2i−1)1 , (2i+1)1 ), ((2i−1)0 , (2i+2)0 , (2i−1)1 , (2i+2)1 ), ((2i)0 , (2i+1)0 , (2i)1 , (2i+1)1 ), ((2i)0 , (2i+2)0 , (2i)1 , (2i+2)1 ), {(2i− 1)0 , (2i)0 }, {(2i − 1)0 , (2i)0 }, {(2i + 1)0 , (2i + 2)0 }, {(2i + 1)0 , (2i + 2)0 }} and Ai = {((2i − 1)0 , (2i)0 , (2i + 1)0 , (2i + 2)0 ), ((2i − 1)0 , (2i)0 , (2i + 2)0 , (2i + 1)0 ), ((2i − 1)1 , (2i + 1)1 , (2i)1 , (2i + 2)1 ), ((2i − 1)0 , (2i + 1)1 , (2i)0 , (2i + 2)1 ), ((2i + 1)0 , (2i − 1)1 , (2i + 2)0 , (2i)1 )}, where xj denotes the ordered pair (x, j). Example 4.3 (n = 13). Let S = {1, 2, 3, 4, 5, 6} and let (S, G, T) be a {3}-GDD of type 23 with groups g1 = {1, 2}, g2 = {3, 4}, and g3 = {5, 6}. Set X = {∞} ∪ (S × Z2 ) and define a collection B of copies of K4 − e as follows: (1) For every i = 1, 2, 3, let ({∞} ∪ (gi × Z2 ), Bi ) be a fivefold K4 − e design of order 5 having a metamorphosis into a maximum packing of 5K5 with 4-cycles with leave the double edge Li = {gi × {0}, gi × {0}} (see Lemma 4.1); put Bi ⊆ B. (2) For every t =(x, y, z) ∈ T , let (V(K2,2,2 ), Bt ) be the K4 −e design in Example 4.2 (from here on we will use the subscript notation ai to denote the ordered pair (a, i)); put Bt ⊆ B. Then (X, B) is a fivefold K4 −e design of order 13. The metamorphosis is the following: for each group g ∈ G use the metamorphosis in (1) and for each triple in T use the metamorphosis in (2). Since (10 , 30 , 11 , 31 ), (10 , 40 , 11 , 41 ), (20 , 30 , 21 , 31 ), (20 , 40 , 21 , 41 ) ∈ C2 (B), we can use these edges in a rearrangement. Combine these edges with the leaves L1 and L2 . This gives a copy of A1 (see above) which can be replaced with Ai to give a maximum packing of 5K13 with 4-cycles with leave L3 . Example 4.4 (n = 23, with all possible leaves). Set X = {∞} ∪ (Z11 × Z2 ) and define a collection B of copies of K4 − e as follows: (1) Let (Z11 × {0}, B0 ) be a threefold K4 − e design of order 11 having a metamorphosis into a maximum packing of 3K11 with 4-cycles with leave L0 (see Example 3.4); put B0 ⊆ B.

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(2) Let (Z11 × {1}, B1 ) be a fourfold K4 − e design of order 11 having a metamorphosis into a maximum packing of 4K11 with 4-cycles with leave the empty set (see Example 3.5); put B1 ⊆ B. (3) Put 2{[i0 , (1+i)0 , (3+i)1 , (5+i)1 ], [(1+i)0 , (3+i)0 , i1 , (2+i)1 ], [(3+i)0 , (6+i)0 , (1+i)1 , (8+i)1 ], [(6+i)0 , i0 , (3+i)1 , (4+i)1 ] | i ∈ Z11 } ⊆ B and {[i1 , (1 + i)1 , (4 + i)0 , (5 + i)0 ], [(1 + i)1 , (3 + i)1 , (7 + i)0 , (2 + i)0 ], [(3 + i)1 , (6 + i)1 , (3 + i)0 , (8 + i)0 ], [i1 , (6 + i)1 , (5 + i)0 , (10 + i)0 ], [∞, i0 , i1 , (4 + i)0 ], [i1 , i0 , (4 + i)0 , ∞], [i1 , (4 + i)1 , i0 , ∞], [∞, (4 + i)1 , (2 + i)0 , (4 + i)0 ]|i ∈ Z11 } ⊆ B. Then (X, B) is a fivefold K4 −e design of order 23. The metamorphosis is the following: use the metamorphosis in (1) and (2); delete the diagonals from the type (3) graphs and rearrange them into the collections of 4-cycles 2{(i0 , (1 + i)0 , (3 + i)0 , (6 + i)0 ) | i ∈ Z11 } and {(i1 , (1 + i)1 , (3 + i)1 , (6 + i)1 ), (∞, i0 , i1 , (4 + i)1 ) | i ∈ Z11 }. Example 4.5 (n = 29). Let S = {1, 2, . . . , 14} and let (S, G, T) be a {3}-GDD of type 27 with groups g1 = {1, 2}, g2 = {3, 4}, . . . , g7 = {13, 14}. Set X = {∞} ∪ (S × Z2 ) and define a collection B of copies of K4 − e as follows: (1) For every i = 1, 2, . . . , 7, let ({∞} ∪ (gi × Z2 ), Bi ) be a fivefold K4 − e design of order 5 having a metamorphosis into a packing of 5K5 with 4-cycles with leave the double edge Li = {gi × {0}, gi × {0}} (see Lemma 4.1); put Bi ⊆ B. (2) For every t = (x, y, z) ∈ T , let (V(K2,2,2 ), Bt ) be the K4 − e design in Example 4.2; put Bt ⊆ B. Then (X, B) is a fivefold K4 −e design of order 29. The metamorphosis is the following: for each group g ∈ G use the metamorphosis in (1) and for each triple in T use the metamorphosis in (2). For i = 2j + 1, with j = 0, 1, 2, since ((2i − 1)0 , (2i + 1)0 , (2i − 1)1 , (2i + 1)1 ), ((2i − 1)0 , (2i + 2)0 , (2i − 1)1 , (2i + 2)1 ), ((2i)0 , (2i + 1)0 , (2i)1 , (2i + 1)1 ), ((2i)0 , (2i + 2)0 , (2i)1 , (2i + 2)1 ) ∈ C2 (B), we can use these edges in a rearrangement. Combining these edges with the leaves Li and Li+1 gives a copy of Ai which can be replaced with Ai to obtain a maximum packing of 5K29 with 4-cycles with leave L7 . We will now give a further example which will be necessary for the subsequent recursive contructions. In this example and in the constructions we will repeatedly use the following theorem. Theorem 4.1 (Sotteau, 1981). Necessary and sufficient conditions for the complete bipartite graph Km,n to be partitioned into 2k-cycles are: (i) m and n are even, (ii) k  m and n, and (iii) 2k|mn. In what follows we will be partitioning Km,n into 4-cycles, so the necessary and sufficient conditions of Sotteau's Theorem reduce to simply m and n are even. Example 4.6 (n = 10 + h, h ∈ {3, 7, 9}, with a hole H of size h, having a metamorphosis into a packing of 5(K10+h \H) with 4-cycles with leave a triangle with exactly one vertex in H). Let (P, ◦) be an idempotent anti-symmetric quasigroup of order 10 + h with a hole of size h. Define B as in Example 4.2. Then (P, ◦) defines a fivefold K4 − e design (P, B) of order 10 + h with a hole of size h. The metamorphosis is the following. D(B) is a copy of K10+h \H. Write K10+h \H = K10 + K10,1 + K10,h−1 and let V(K10 ) = Z9 ∪ {∞} and H = {∞0 , ∞1 , ∞2 , . . . , ∞h−1 }. Partition the edges of K10 ∪ K10,1 , where the star K10,1 has ∞0 as center, into the collection of 4-cycles {(i, 2 + i, 5 + i, 4 + i) | 0  i  4, i ∈ Z9 } ∪ {(i, ∞, 5 + i, 4 + i) | 5  i  8, i ∈ Z9 } ∪ {(i, 2 + i, 5 + i, ∞0 ) | 5  i  8, i ∈ Z9 } and the triangle (∞0 , ∞, 0). Finally, by using Sotteau's Theorem partition the edges of K10,h−1 into 4-cycles. With the previous examples in hand we can now give six recursive constructions which handle all of the remaining cases. The 40k + 7 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k), with holes H = {h1 , h2 , . . . , h4k } of size 2. Set X = {∞0 , ∞1 , . . . , ∞6 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , . . . , ∞6 }, B ) be a fivefold K4 − e design of order 7 having a metamorphosis into a maximum packing of K7 with 4-cycles with leave L (see Remark 4.1 and Example 4.1); put B ⊆ B. (2) For each hi ∈ H, let ({∞0 , ∞1 , . . . , ∞6 }∪(hi ×{1, 2, 3, 4, 5}), hi ) be a fivefold K4 −e design of order 17 with hole {∞0 , ∞1 , . . . , ∞6 } having a metamorphosis into a collection of 4-cycles with leave the triangle (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (3) If x and y belong to different holes put 5[xi , yi , (x ◦ y)i+3 , (x ◦ y)i+5 ] in B. Then (X, B) is fivefold K4 −e design of order 40k+7. The metamorphosis is the following. (i) Use the metamorphosis in (1). (ii) Delete all edges of the form {x1 , y1 }, where x and y belong to different holes of H. Partition H into k sets of size 4, say  = {1 , 2 , . . . , k }. For each j ∈ , the deleted edges {x1 , y1 }, where x and y belong to different holes of j , plus the four triangles (∞, a, b), {a, b} ∈ j , form a copy of K9 . Partition these edges into 4-cycles. Delete all edges of the form {xi , yi }, i = 2, 3, 4, 5, where x and y belong to different holes of H. These edges can be reassembled into 4-cycles using Sotteau's Theorem (Theorem 4.1). Then (X, C1 (B)∪C2 (B), L) is a maximum packing of K40k+7 with 4-cycles with all possible leaves. The leaves come from (1).

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The 40k + 23 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k) + 4, with holes H = {h1 , h2 , . . . , h4k+1 } all of size 2 with the exception of h1 of size 4. Set X = {∞0 , ∞1 , ∞2 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , ∞2 } ∪ (h1 × {1, 2, 3, 4, 5}), h1 ) be a fivefold K4 − e design of order 23 having a metamorphosis into a maximum packing with 4-cycles with leave L (see Example 4.4); put h1 ⊆ B. (2) For each hi ∈ H, i = 2, 3, . . . , 4k + 1, let ({∞0 , ∞1 , ∞2 } ∪ (hi × {1, 2, 3, 4, 5}), hi ) be a fivefold K4 − e design of order 13 with hole {∞0 , ∞1 , ∞2 } having a metamorphosis into a collection of 4-cycles with leave the triangle (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (3) If x and y belong to different holes of H, the same as (3) in the 40k + 7 construction. Then (X, B) is fivefold K4 − e design of order 40k + 23. The metamorphosis is the following: use the metamorphosis in (1) and (2), delete the edges {xi , yi }, x and y in different holes of H, and reassemble these edges as in the 40 + 7 construction. Then (X, C1 (B) ∪ C2 (B), L) is a maximum packing of K40k+23 with 4-cycles with all possible leaves. The leaves come from (1). The 40k + 39 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k + 3), with holes H = {h1 , h2 , . . . , h4k+3 } of size 2. Set X = {∞0 , ∞1 , . . . , ∞8 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , . . . , ∞8 } ∪ (h1 × {1, 2, 3, 4, 5}), B1 ) be a threefold K4 − e design of order 19 having a metamorphosis into a maximum packing with leave L (see Section 3); put B1 ⊆ B. (2) Let ({∞0 , ∞1 , . . . , ∞8 } ∪ (h1 × {1, 2, 3, 4, 5}), B1 ) be a twofold K4 − e design of order 19 having a metamorphosis into a maximum packing with leave {{∞0 , ∞1 }, {∞0 , ∞1 }} (see Section 3); put h1 ⊆ B. (3) For i = 2, 3, let ({∞0 , ∞1 , . . . , ∞8 } ∪ (hi × {1, 2, 3, 4, 5}), hi ) be a fivefold K4 − e design of order 19 with hole {∞0 , ∞1 , . . . , ∞8 } having a metamorphosis into a collection of 4-cycles with leave a triangle with exactly one vertex in {∞0 , ∞1 , . . . , ∞8 } (see Example 4.6); put hi ⊆ B. (4) For each hi ∈ H, i = 4, 5, . . . , 4k + 3, let ({∞0 , ∞1 , . . . , ∞8 } ∪ (hi × {1, 2, 3, 4, 5})) be a fivefold K4 − e design of order 19 with hole {∞0 , ∞1 , . . . , ∞8 } having a metamorphosis into a collection of 4-cycles with leave (∞1 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (5) If x and y belong to different holes of H, the same as (3) in the 40k + 7 construction. Then (X, B) is fivefold K4 − e design of order 40k + 39. The metamorphosis is the following. Use the metamorphosis in (1), (2), and (4). In (3) use the metamorphosis with leave (∞0 , 31 , 41 ) and (∞1 , 51 , 61 ), where h2 ={3, 4} and h3 ={5, 6}. Delete the edges {xi , yi }, x and y in different holes of H, and reassemble these edges as follows. Combine the leaves {{∞0 , ∞1 }, {∞0 , ∞1 }}, (∞0 , 31 , 41 ), and (∞1 , 51 , 61 ) with {31 , 51 }, {31 , 61 }, {41 , 51 }, {41 , 61 } and partition these deleted edges into the 4-cycles (∞1 , ∞0 , 31 , 61 ), (∞1 , ∞0 , 41 , 51 ), (31 , 41 , 61 , 51 ). Combine the 4 leaves in (4) as in the 40k + 7 construction. Finally, partition all type (5) edges not already used into 4-cycles. Then (X, C1 (B) ∪ C2 (B), L) is a maximum packing of K40k+39 with 4-cycles with all possible leaves. The leaves come from (1). The 40k + 13 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k + 1), with holes H = {h1 , h2 , . . . , h4k+1 } of size 2. Set X = {∞0 , ∞1 , ∞2 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , ∞2 } ∪ (h1 × {1, 2, 3, 4, 5}), h1 ) be a fivefold K4 − e design of order 13 having a metamorphosis into a maximum packing with 4-cycles with leave the double edge L (see Example 4.3); put h1 ⊆ B. (2) For each hi ∈ H, i = 2, 3, . . . , 4k + 1, let ({∞0 , ∞1 , ∞2 } ∪ (hi × {1, 2, 3, 4, 5}), hi ) be a fivefold K4 − e design of order 13 with hole {∞0 , ∞1 , ∞2 } having a metamorphosis into a collection of 4-cycles with leave (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (3) If x and y belong to different holes of H, the same as (3) in the 40k + 7 construction. Then (X, B) is fivefold K4 − e design of order 40k + 13. The metamorphosis is the following: use the metamorphosis in (1) and (2), delete the edges {xi , yi }, x and y in different holes of H, and reassemble these edges as in the 40 + 7 construction. Then (X, C1 (B) ∪ C2 (B), L) is a maximum packing of K40k+13 with 4-cycles. The leave comes from (1). The 40k + 29 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k) + 4, with holes H = {h1 , h2 , . . . , h4k+1 } all of size 2 with the exception of h1 of size 4. Set X = {∞0 , ∞1 , . . . , ∞8 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , . . . , ∞8 }∪(h1 ×{1, 2, 3, 4, 5}), h1 ) be a fivefold K4 −e design of order 29 having a metamorphosis into a maximum packing with leave the double edge L (see Example 4.5); put h1 ⊆ B. (2) For each hi ∈ H, i = 2, 3, . . . , 4k + 1, let ({∞0 , ∞1 , . . . , ∞8 } ∪ (hi × {1, 2, 3, 4, 5}) be a fivefold K4 − e design of order 19 with hole {∞0 , ∞1 , . . . , ∞8 } having a metamorphosis into a collection of 4-cycles with leave (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (3) If x and y belong to different holes of H, the same as (3) in the 40k + 7 construction.

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Then (X, B) is fivefold K4 − e design of order 40k + 29. The metamorphosis is the following: use the metamorphosis in (1) and (2), delete the edges {xi , yi }, x and y in different holes of H, and reassemble these edges as in the 40 + 7 construction. Then (X, C1 (B) ∪ C2 (B), L) is a maximum packing of K40k+29 with 4-cycles. The leave come from (1). The 40k + 37 construction: Let (S, ◦) be a commutative quasigroup of order 2(4k + 3), with holes H = {h1 , h2 , . . . , h4k+3 } of size 2. Set X = {∞0 , ∞1 , . . . , ∞6 } ∪ (S × {1, 2, 3, 4, 5}) and define a collection B of copies of K4 − e as follows: (1) Let ({∞0 , ∞1 , . . . , ∞6 } ∪ (h1 × {1, 2, 3, 4, 5}), h1 ) be a fivefold K4 − e design of order 17 having a metamorphosis into a fivefold 4-cycle system (see Lemma 4.2); put h1 ⊆ B. (2) For i = 2, 3, let ({∞0 , ∞1 , . . . , ∞6 } ∪ (hi × {1, 2, 3, 4, 5}), hi ) be a fivefold K4 − e design of order 17 with hole {∞0 , ∞1 , . . . , ∞6 } having a metamorphosis into a collection of 4-cycles with leave (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (3) For each hi ∈ H, i = 4, 5, . . . , 4k + 3, let ({∞0 , ∞1 , . . . , ∞6 } ∪ (hi × {1, 2, 3, 4, 5}), hi ) be a fivefold K4 − e design of order 17 with hole {∞0 , ∞1 , . . . , ∞6 } having a metamorphosis into a collection of 4-cycles with leave (∞0 , x1 , y1 ), where hi = {x, y} (see Example 4.6); put hi ⊆ B. (4) If x and y belong to different holes of H, the same as (3) in the 40k + 7 construction. Then (X, B) is fivefold K4 − e design of order 40k + 37. The metamorphosis is the following: use the metamorphosis in (1), (2), and (3), delete the edges {xi , yi }, x and y in different holes of H, and reassemble these edges as follows. Combine the leaves (∞0 , 31 , 41 ) and (∞0 , 51 , 61 ), where h2 = {3, 4} and h3 = {5, 6}, with all of the edges between the holes h2 and h3 and partition these deleted edges into the 3 4-cycles (∞0 , 51 , 31 , 61 ), (∞0 , 31 , 61 , 41 ), (31 , 51 , 61 , 41 ), the 3-times repeated 4-cycle (51 , 31 , 61 , 41 ), and the double edge {{41 , 51 }, {41 , 51 }}. Combine the 4 leaves in (3) as in the 40k + 7 construction. Finally, partition all type (3) edges not already used into 4-cycles. Then (X, C1 (B) ∪ C2 (B), L) is a maximum packing of K40k+37 with 4-cycles, where L = {{41 , 51 }, {41 , 51 }}. Combining all of the above results gives the following lemma. Lemma 4.3. For every n ≡ 7, 13, 23, 29, 37, 39 (mod 40) there exists a fivefold K4 − e design of order n having a metamorphosis into a maximum packing of 5Kn with 4-cycles with all possible leaves. Combining Lemmas 4.1--4.3 and Remark 4.1 gives the following result. Lemma 4.4. For every n  4 there exists a fivefold K4 − e design of order n having a metamorphosis into a maximum packing of 5Kn with 4-cycles with all possible leaves. Let now (P, B) be a fivefold K4 − e design of order n  4 obtained by the Folk construction. If we replace D(B) with a maximum packing of Kn with 4-cycles we have a packing of 5Kn with leave as in Fig. 1. By combining two, three, or four copies of such designs and by arguments similar to Lemmas 3.1--3.3 we have the following lemma. Lemma 4.5. For every  = 10, 15, 20 and for every n  4 there exists a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves. Finally, for any value of  ≡ 0 (mod 5)  25, write  = 20k + h where h = 5, 10, 15 and paste together k solutions for  = 20 with a solution for  = h. 5. Summary For any value of  ≡ / 0 (mod 5) > 1, write  = 4k + h where 2  h  5 and paste together k solutions for  = 4 with a solution for

 = h.

Theorem 5.1. For every  and for every admissible value of n there exists a -fold K4 − e design of order n having a metamorphosis into a maximum packing of Kn with 4-cycles with all possible leaves. References Gionfriddo, M., Lindner, C.C., Rodger, C.A., 1991. 2-Colouring K4 − e designs. Australasian J. Combin. 3, 211--229. Lindner, C.C., Tripodi, A., 2005. The metamorphosis of K4 − e designs into maximum packings of Kn with 4-cycles. Ars Combin. 75, 333--349.  ) into cycles (circuits) of length 2k. J. Combin. Theory Ser. B 30, 75--81. Sotteau, D., 1981. Decompositions of Km,n (Km,n