Maximum cyclic 4-cycle packings of the complete multipartite graph

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(1)J Comb Optim (2007) 14: 365–382 DOI 10.1007/s10878-007-9048-6. Maximum cyclic 4-cycle packings of the complete multipartite graph Shung-Liang Wu · Hung-Lin Fu. Published online: 31 March 2007 © Springer Science+Business Media, LLC 2007. Abstract A graph G is said to be m-sufficient if m is not exceeding the order of G, each vertex of G is of even degree, and the number of edges in G is a multiple of m. A complete multipartite graph is balanced if each of its partite sets has the same size. In this paper it is proved that the complete multipartite graph G can be decomposed into 4-cycles cyclically if and only if G is balanced and 4-sufficient. Moreover, the problem of finding a maximum cyclic packing of the complete multipartite graph with 4-cycles are also presented. Keywords Complete multipartite graph · Cyclic · Cycle system · Cycle packing · 4-cycle. 1 Introduction An m-cycle, written (c0 , c1 , . . . , cm−1 ), consists of m distinct vertices c0 , c1 , . . . , cm−1 , and m edges {ci , ci+1 }, 0 ≤ i ≤ m − 2, and {c0 , cm−1 }. An m-cycle system of a simple graph G is a set C of edge disjoint m-cycles which partition the edge set of G. If G is a complete graph on v vertices, it is known as an m-cycle system of order v. The obvious necessary conditions for the existence of an m-cycle system of a graph G are that the value of m is not exceeding the order of G, m divides the number of edges in G, and the degree of each vertex in G is even. A graph G is called m-sufficient if the necessary conditions are met. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. S.-L. Wu National United University, Miaoli, Taiwan H.-L. Fu () Department of Applied Mathematics, National Chaio Tung University, Hsin Chu, Taiwan e-mail: [email protected].

(2) 366. J Comb Optim (2007) 14: 365–382. A graph G is said to be a complete r-partite graph (r > 1) if its vertex set V can be partitioned into r disjoint non-empty sets V1 , . . . , Vr (called partite sets) such that there exists exactly one edge between each pair of vertices from different partite sets. If |Vi | = ni for 1 ≤ i ≤ r, the complete r-partite graph is denoted by Kn1 ,...,nr . In particular, if n1 = · · · = nr = k (>1), it is called balanced and the graph will be simply denoted by Kr(k) . The graph decomposition problem has attracted many researchers, and it serves as useful models for a range of applications such as: serology (Ree 1967), synchronous optical network ring (Colbourn and Wan 2001; Wan 1999), and DNA library screening (Mutoh et al. 2003). The study of m-cycle systems of the complete graph has been one of the most interesting problems in graph decomposition. The existence question for m-cycle systems of the complete graph has been completely settled by Alspach and Gavlas (2001) in the case of m odd and by Šajna (2002) in the even case. The problem of finding the existence of m-cycle systems of the complete r-partite graph has also been considered by a number of researchers. The case when r = 2 and m is even was completely solved by Sotteau (1981). Cavenagh (1998) proved that there exists a k-cycle system of K3(m) if and only if k ≤ 3m and k divides 3m2 . Billington (1999) gave the necessary and sufficient conditions for the existence of a decomposition of any complete tripartite graph into specific numbers of 3-cycles and 4-cycles. Hoffman et al. (1989) proved that if both r and m are odd then there exists an m-cycle system of Kr(m) . The necessary and sufficient conditions to partition the same graph into Hamiltonian cycles are given by Laskar (1978). The existence for 5-cycle system of the complete tripartite graph has been considered by Mahmoodian and Mirzakhani (1995), Cavenagh and Billington (2002), and Cavenagh (2002). Moreover, necessary and sufficient conditions are also given (Cavenagh and Billington 2000) for the existence of m-cycle systems of the complete r-partite graph with m = 4, 6, and 8. An m-cycle packing of a graph G is a set P of edge disjoint m-cycles in G. The leave of an m-cycle packing of G is the set of edges in G that occur in no m-cycle in P . An m-cycle packing P of G is maximum if |P | ≥ |P | for all other m-cycle packings P of G. Obviously, a maximum packing will have a minimum leave, and an m-cycle system of G is an m-cycle packing of G for which the leave is empty. Not much work has been done on packing complete r-partite graphs with cycles. For 3- and 6-cycles, maximum packings in Kr(k) are respectively dealt with in (Billington and Lindner 1996; Fu and Huang 2004). In (Billington et al. 2001), the problem of finding a maximum packing of the complete r-partite graph with 4-cycles is completely solved. A natural generalization to determine a maximum packing of the λ-fold complete r-partite graph appears in (Billington et al. 2005). Let C = (c0 , c1 , . . . , cm−1 ) be an m-cycle. An m-cycle system (packing) of a graph G, C(P ), is said to be cyclic if V (G) = Zv and we have (c0 + 1, c1 + 1, . . . , cm−1 + 1) (mod v) ∈ C(P ) whenever (c0 , c1 , . . . , cm−1 ) ∈ C(P ). The existence question for cyclic m-cycle systems of order v has been completely solved for m = 3 (Peltesohn 1938), 5 and 7 (Rosa 1966b). For m even and v ≡ 1 (mod 2m), cyclic m-cycle systems of order v are proved for m ≡ 0 (mod 4) (Kotzig 1965) and for m ≡ 2 (mod 4) (Rosa 1966a). Recently, it is shown in (Buratti and Del Fra 2003; Bryant et al. 2003; Fu and Wu 2004) that for each pair of integers (m, n), there.

(3) J Comb Optim (2007) 14: 365–382. 367. Table 1 Best possible leaves of a maximum cyclic packing of K r(k) with 4-cycles r. k 0. 1. 2. 3. 4. 5. 6. 7. 0. -. F. -. F. -. F. -. F. 1. -. -. -. -. -. -. -. -. 2. -. F. -. F ∪H. -. F ∪ 2H. -. F ∪ C∗. 3. -. H. -. 3H. -. H. -. 3H. 4. -. F. -. F ∪C. -. F. -. F ∪C. 5. -. 2H. -. 2H. -. 2H. -. 2H. 6. -. F. -. F ∪H. -. F ∪ 2H. -. F ∪C. 7. -. 3H. -. H. -. 3H. -. H. ∗ When r = 2 and k ≡ 7 (mod 8), the leave is the union of a 1-factor and 3 Hamiltonian cycles. exists a cyclic m-cycle system of order 2mn + 1, and in particular, for each odd prime p, there exists a cyclic p-cycle system (Buratti and Del Fra 2003; Fu and Wu 2004). For v ≡ m (mod 2m), cyclic m-cycle systems of order v are presented for m ∈ M (Buratti and Del Fra 2004), where M = {p α | p is prime, α > 1} ∪ {15}, and in (Vietri 2004) for m ∈ M. More recently, the present authors (Wu and Fu 2006) prove that for m = 3, 4, . . . , 32, there exists a cyclic m-cycle system and for p a prime power, there exists a cyclic 2p-cycle system. In this paper, we shall focus on maximum cyclic 4-cycle packings of Kr(k) with leave and the main result is listed in Table 1, where the values of r and k are reduced modulo 8 and the symbols -, iH, C, and F denote respectively the empty set, i Hamiltonian cycles, 2 (rk/2)-cycles, and a 1-factor. In Sect. 2, we will give the essential definitions and preliminaries. In Sect. 3, a cyclic 4-cycle system of Kr(k) will be presented, and in Sects. 4 and 5, maximum cyclic 4-cycle packings of Kr(k) with leave and with rk odd or even will be respectively given.. 2 Definitions and preliminaries Assume {a, b} to be any edge of G with V (G) ∈ Zv . We shall use ±|a − b| to denote the difference of the edge {a, b} in G. The number of distinct differences in a graph G defined on Zv is called the weight of G, denoted by W (G). Let C = (c0 , c1 , . . . , cm−1 ) be an m-cycle of G and let C + i = (c0 + i, c1 + i, . . . , cm−1 + i) (mod v), where i ∈ Zv . A cycle orbit O of C is a collection of distinct m-cycles in {C + i | i ∈ Zv }. The length of a cycle orbit is its cardinality, i.e., the minimum positive integer k such that C + k = C. A base cycle of a cycle orbit O is a cycle C ∈ O that is chosen arbitrarily. For the convenience of notation, we write a cycle k-orbit for a cycle orbit of length k. A cycle v-orbit of C on G is said to be full and otherwise short. Given a subset Ω of Zv − {0} with Ω = −Ω, the circulant graph X(Zv , Ω) of order v is the Cayley graph Cay[Zv ; Ω], that is, the graph with vertex set Zv and all.

(4) 368. J Comb Optim (2007) 14: 365–382. possible edges of the form {x, x + w} with w ∈ Ω. The set Ω is called the connection set and its size is the degree of X(Zv , Ω). We first introduce a necessary condition for the existence of a cyclic m-cycle system of a graph. Lemma 2.1 If there is a cyclic m-cycle system of a graph G, then G is 2r-regular for some positive integer r. Proof For i = 1, . . . , p with p ≥ 1, let Oi be a cycle ki -orbit of Ci in the cyclic mcycle system and let Ci be the base cycle of Oi with weight wi . Note that the graph induced by the edges having the same difference is a spanning 2-regular subgraph of G. Thus, the union of the cycles Ci , Ci + 1, . . . , Ci + (ki − 1) forms a spanning 2wi regular subgraph of G. This means that each cycle ki -orbit Oi (1 ≤ i ≤ p) is exactly p a spanning 2wi -regular subgraph of G. It follows that the graph G is (2 i=1 wi )regular. Remark that the graph G in Lemma 2.1 is precisely a circulant graph. It is clear from Lemma 2.1 that if there exists a cyclic m-cycle system of the complete r-partite graph Kn1 ,...,nr , then Kn1 ,...,nr is balanced, namely, n1 = · · · = nr = k for some integer k (>1). A necessary condition for the existence of a cyclic m-cycle system of Kr(k) is that any partite set in Kr(k) is the subgroup rZk = {0, r, . . . , (k − 1)r} of Zrk or its coset. For i = 0, . . . , r − 1, let Vi denote the ith partite set of Kr(k) . Throughout this paper we will assume the ith partite set of Kr(k) to be Vi = {i, i + r, . . . , i + (k − 1)r} for i = 0, . . . , r − 1. Note that the set of distinct differences of edges in Kr(k) is Zrk \ ± {0, r, . . . , k/2 r}. For an m-cycle C with V (C) ∈ Zv , the necessary condition for the sum of absolute differences of edges in C is given as follows: Lemma 2.2 Let C = (c0 , c1 , . . . , cm−1 ) be an m-cycle with ci ∈ Zv where 0 ≤ i ≤ m − 1 and v is any positive integer. Then the sum of absolute differences of edges in C is even. Proof The proof follows immediately from the fact that m i=1. |ci − ci−1 | ≡. m (ci − ci−1 ) ≡ 0 (mod 2). i=1. . The following consequences can be obtained by simple observations. Lemma 2.3 If C is an m-cycle with weight p in a cyclic m-cycle system of Kr(k) , then m is a multiple of p. Consequently, if m = pq, then the value of q is a common divisor of m and rk..

(5) J Comb Optim (2007) 14: 365–382. 369. Lemma 2.4 Suppose Ω = ±{b} with b ∈ Z v/2 and let k = v/gcd(v, b). Then the circulant graph X(Zv , Ω) is the union of v/k edge-disjoint k-cycles. If gcd(v, b) = 1, then X(Zv , Ω) is exactly a Hamiltonian cycle in Kv and if b = v/m, then X(Zv , Ω) is the union of b edge-disjoint m-cycles. ∗ Lemma 2.5 Let ai (1 ≤ i ≤ 4) be distinct elements in Z v/2 = Z v/2 \{0}. If Ω = ±{a1 , a2 , a3 , a4 } with a1 + a2 = a3 + a4 , then there exists a cyclic 4-cycle system of X(Zv , Ω).. Proof The base cycle is (0, a1 , a1 + a2 , a3 ).. . Lemma 2.6 If Ω = ±{a1 , a2 } with a1 = a2 and a1 + a2 = rk/2, then there exists a cyclic 4-cycle system of X(Zrk , Ω). Proof Choose (0, a1 , rk/2, rk/2 + a1 ) as the base cycle.. . Given a positive integer m = pq, an m-cycle C in Kr(k) with weight p has index rk/q if for each edge {s, t} in C, the edges {s + i · rk/q, t + i · rk/q} (mod rk) with i ∈ Zq are also in C. For instance, the 15-cycle C = (0, 1, 5, 7, 12, 25, 26, 30, 32, 37, 50, 51, 55, 57, 62) in K5(15) with weight 5 (differences ±1, ±2, ±4, ±5, and ±13) has index 25. The following consequence will be the crucial tool for constructing a cycle orbit in a cyclic m-cycle system of Kr(k) . The similar results about 1-rotational m-cycle system of the complete graph can also be found in (Buratti 2003, 2004) and so we omit the details. Proposition 2.7 Let m = pq. Then there exists an m-cycle C = (c0 , c1 , . . . , cm−1 ) in Kr(k) with weight p and index rk/q if and only if each of the following conditions is satisfied: (1) (2) (3) (4). For 0 ≤ i = j ≤ p − 1, ci ≡ cj (mod rk/q); The differences of the edges {ci , ci−1 } (1 ≤ i ≤ p) are all distinct; cp − c0 = t · rk/q, where gcd(t, q) = 1; and cip+j = cj + i · t · rk/q where 0 ≤ j ≤ p − 1 and 0 ≤ i ≤ q − 1.. It should be noticed that in Proposition 2.7, the m-cycle C can be viewed as a base cycle and the set {C + i | i ∈ Zrk/q } forms a cycle (rk/q)-orbit of C in Kr(k) . To simplify, C will be denoted by C = [c0 = 0, c1 , . . . , cp−1 ]t·rk/q , and we denote the set of partial differences ±{(ci − ci−1 ) | 1 ≤ i ≤ p} of C by ∂C. Consider, for instance, the 8-cycle C = (0, 15, 14, 29, 28, 43, 42, 1) = [0, 15]14 in K7(8) with weight 2 (i.e., ∂C = ±{1, 15}) and index 14, and the set {C, C + 1, . . . , C + 13} forms a cycle 14-orbit of C in K7(8) . Given a set D = {C1 , . . . , Ct } of m-cycles, the list of differences from D is defined as the union of the multisets ∂C1 , . . . , ∂Ct , i.e., ∂D = ti=1 ∂Ci . The next result is simple but important and will be used later. Theorem 2.8 A set D of m-cycles with vertices in Zrk is a set of base cycles of a cyclic m-cycle system of Kr(k) if and only if ∂D = Zrk \ ± {0, r, . . . , k/2 r}..

(6) 370. J Comb Optim (2007) 14: 365–382. 3 Cyclic 4-cycle systems Theorem 3.1 The complete multipartite graph G can be decomposed into 4-cycles cyclically if and only if G is balanced and 4-sufficient. Proof (Necessity) Since G can be decomposed cyclically, it follows from Lemma 2.1 that G must be a regular graph. Hence, G is a balanced complete multipartite graph Kr(k) for some positive integers r and k. Now, if k is even, then clearly the degree of every vertex of G is even and 4||E(G)|. On the other hand, if k is odd, then r must be odd in order that each vertex of G is of even degree. Moreover, 4||E(G)| implies that r ≡ 1 (mod 8). Therefore, we have that G is 4-sufficient. (Sufficiency) By virtue of Theorem 2.8, it suffices to prove that there is a set D of base cycles in Kr(k) so that ∂D = Zrk \ ± {0, r, . . . , k/2 r}. We break the proof into two cases depending on whether k is even or odd. Case 1. k is even. (1) k ≡ 0 (mod 4), say k = 4p. For i ∈ Zp and j ∈ Zr∗ , let Ci,j = [0, j + ir]2pr . Clearly, ∂Ci,j = ±{j + ir, (2p − i)r − j }. Therefore, {Ci,j } is a set of base cycles we need. (2) k ≡ 2 (mod 4) and r ≡ 0 (mod 2), say k = 4p + 2. Again, for i ∈ Zp and j ∈ Zr∗ , let Ci,j = [0, j + ir](2p+1)r . Moreover, let C = ∗ . (0, (2p + 1)r/2, (2p + 1)r, 3(2p + 1)r/2), and Ct = [0, t + pr](2p+1)r for t ∈ Zr/2 Then ∂C ∪ {∂Ct } = ±{1 + pr, 2 + pr, . . . , r − 1 + pr}. Hence, {Ci,j } ∪ {C} ∪ {Ct } consists of a set of base cycles. (3) k ≡ 2 (mod 4) and r ≡ 1 (mod 2), say k = 4p + 2. For i ∈ Zp and j ∈ Zr∗ , let Ci,j = [0, j + ir](2p+1)r and Ct = [0, t + pr](2p+1)r for ∗ t ∈ Z(r+1)/2 . Since (∪∂Ci,j ) ∪ (∪∂Ct ) = Zrk \ ± {0, r, . . . , (2p + 1)r}, {Ci,j } ∪ {Ct } forms a set of base cycles. Case 2. k is odd and r ≡ 1 (mod 8), say k = 2h + 1 and r = 8q + 1. For i ∈ Zh and j ∈ Z2q , let Ci,j = (0, 4j + 1 + ir, 8j + 5 + 2ir, 4j + 2 + ir), and let Ct = (0, 4t + 1 + hr, 8t + 5 + 2hr, 4t + 2 + hr) for t ∈ Zq . Since ∂Ci,j = ±{4j + 1 + ir, 4j + 2 + ir, 4j + 3 + ir, 4j + 4 + ir} and ∂Ct = ±{4t + 1 + hr, 4t + 2 + hr, 4t + 3 + hr, 4t + 4 + hr}, we have a set {Ci,j } ∪ {Ct } of base cycles for the cycle system. Now, we are ready for the packings with cyclic 4-cycles. We shall classify the maximum cyclic m-cycle packings of Kr(k) with leave into two cases: Odd and Even according as the value of order of Kr(k) is odd or even. 4 Maximum cyclic 4-cycle packings of Kr(k) of odd order Since there exists a cyclic 4-cycle system of Kr(k) whenever k is odd and r ≡ 1 (mod 8), here we consider the remaining cases. That is, when k is odd and r ≡ 3, 5, or 7 (mod 8), no cyclic 4-cycle system of Kr(k) exists. The following consequence indicates the possible leave of a maximum cyclic 4cycle packing of Kr(k) and will be utilized repeatedly in this section. Given a maximum cyclic 4-cycle packing of Kr(k) , P , let D(P ) be the set of distinct differences in P ..

(7) J Comb Optim (2007) 14: 365–382. 371. Lemma 4.1 Suppose that rk ≡ 1 (mod 2) and W (Kr(k) ) ≡ i (mod 4) with i ∈ Z4∗ and let P be a maximum cyclic 4-cycle packing ofKr(k) . Then the leave of a maximum cyclic 4-cycle packing of Kr(k) is the circulant graph X(Zrk , Ω) with Ω = Zrk \ ± {0, r, . . . , k/2 r}\D(P ). Proof Since the value of rk is odd, each cycle orbit in the maximum cyclic 4cycle packing of Kr(k) , P , must be full, and since W (Kr(k) ) ≡ i (mod 4) with i ∈ Z4∗ , it implies that there are exactly i distinct differences not occurring in P . It follows that the leave is precisely the circulant graph X(Zrk , Ω) with Ω = Zrk \ ± {0, r, . . . , k/2 r}\D(P ). Throughout this paper whenever we say that a circulant graph X(Zrk , ±{a}) is a Hamiltonian cycle of Kr(k) , it implies that gcd(rk, a) = 1. Given a connection set Ω = ±{a1 , . . . , at }, let Ω ⊕ i = ±{a1 + i, . . . , at + i}. We are now in a position to prove our main result with odd order, which is divided into the following five propositions. Proposition 4.2 If r ≡ 3 (mod 8) and k ≡ 3 (mod 4), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave 3 Hamiltonian cycles. Proof Since W (Kr(k) ) ≡ 3 (mod 4), by Lemma 4.1, the leave is a circulant graph X(Zrk , Ω) with |Ω| = 3. Let Ω1∗ = ±{1}, Ω2∗ = ±{2}, and Ω3∗ = ±{(rk − 1)/2}. Then the circulant graph X(Zrk , Ω) is the union of X(Zrk , Ωi∗ ) for i = 1, 2, 3. Note that by Lemma 2.4, the circulant graphs X(Zrk , Ωi∗ ) (1 ≤ i ≤ 3) are all Hamiltonian cycles in Kr(k) . The remaining proof are split into two cases according to whether r = 3 or r > 3. Let r = 8t + 3 and k = 4s + 3. Case 1. r = 3. Let Ωi = ±{4, 5, 7, 8} ⊕ 6i for i = 0, . . . , s − 1. Note that by Lemma 2.5, there exists a cyclic 4-cycle system of X(Zrk , Ωi ) for each i. It is easy to check that the union of the circulant graphs X(Zrk , Ωi ) (0 ≤ i ≤ s − 1) consists of a maximum cyclic 4-cycle packing of Kr(k) . Case 2. r > 3. The connection sets are given as the following: Ωi = ±{r + 1, r + 2, 2r + 1, 2r + 2} ⊕ 2ir, Ωi,j = ±{3, 4, 5, 6} ⊕ 4i ⊕ rj,. i = 0, . . . , s − 1;. i = 0, . . . , 2t − 1 and j = 0, . . . , 2s;. Ωi = ±{2sr + r + 1, 2sr + r + 2, 2sr + r + 3, 2sr + r + 4} ⊕ 4i, i = 0, . . . , t − 1. Again, a routine verification shows that the union of the circulant graphs X(Zrk , Ωi ), X(Zv , Ωi,j ), and X(Zv , Ωi ) forms a maximum cyclic 4-cycle packing of Kr(k) . Proposition 4.3 If r ≡ 3 (mod 8) and k ≡ 1 (mod 4), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a Hamiltonian cycle..

(8) 372. J Comb Optim (2007) 14: 365–382. Proof Analogously, the leave is a Hamiltonian cycle, i.e., the circulant graph X(Zrk , ±{(rk −1)/2}). Also, we divide the proof into two cases according to whether r = 3 or r > 3. Let r = 8t + 3 and k = 4s + 1. Case 1. r = 3. Let Ωi = ±{1, 2, 4, 5} ⊕ 6i for i = 0, . . . , s − 1 and the union of the circulant graphs X(Zrk , Ωi ) (0 ≤ i ≤ s − 1) is a maximum cyclic 4-cycle packing of Kr(k) . Case 2. r > 3. The connection sets are defined by Ωi = ±{1, 2, r + 1, r + 2} ⊕ 2ir, Ωi,j = ±{3, 4, 5, 6} ⊕ 4i ⊕ rj,. i = 0, . . . , s − 1; i = 0, . . . , 2t − 1 and j = 0, . . . , 2s − 1;. Ωi = ±{2sr + 1, 2sr + 2, 2sr + 3, 2sr + 4} ⊕ 4i,. i = 0, . . . , t − 1.. An easy computation shows that the union of the circulant graphs X(Zrk , Ωi ), X(Zv , Ωi,j ), and X(Zv , Ωi ) forms a maximum cyclic 4-cycle packing of Kr(k) . Proposition 4.4 If r ≡ 5 (mod 8) and k ≡ 1 (mod 2), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave 2 Hamiltonian cycles. Proof The Hamiltonian cycles are the circulant graphs X(Zrk , Ωi∗ = ±{i}) for i = 1, 2. Let r = 8t + 5. Then, by a similar argument, it suffices to provide the connection sets which are the following: Ω = ±{3, 4, k/2 r + 1, k/2 r + 2}; Ωi = ±{5, 6, 7, 8} ⊕ 4i,. i = 0, . . . , 2t − 1;. Ωi,j = ±{r + 1, r + 2, r + 3, r + 4} ⊕ 4i ⊕ rj, i = 0, . . . , 2t and j = 0, . . . , (k − 5)/2; Ωi. = ±{ k/2 r + 3, k/2 r + 4, k/2 r + 5, k/2 r + 6} ⊕ 4i,. i = 0, . . . , t − 1.. . Proposition 4.5 If r ≡ 7 (mod 8) and k ≡ 1 (mod 4), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave 3 Hamiltonian cycles. Proof For i = 1, 2, 3, the circulant graphs X(Zrk , Ωi∗ ) with Ω1∗ = ±{1}, Ω2∗ = ±{2}, and Ω3∗ = ±{ rk/2 } are the Hamiltonian cycles. Let r = 8t + 7 and k = 4s + 1. Then, with the connection sets defined below, we have the proof. Ωi = ±{r + 1, r + 2, 2r + 1, 2r + 2} ⊕ 2ir, Ωi,j = ±{3, 4, 5, 6} ⊕ 4i ⊕ rj, Ωi. i = 0, . . . , s − 1;. i = 0, . . . , 2t and j = 0, . . . , 2s − 1;. = ±{2sr + 3, 2sr + 4, 2sr + 5, 2sr + 6} ⊕ 4i,. i = 0, . . . , t − 1.. . Proposition 4.6 If r ≡ 7 (mod 8) and k ≡ 3 (mod 4), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a Hamiltonian cycle..

(9) J Comb Optim (2007) 14: 365–382. 373. Proof The Hamiltonian cycle is the circulant graph X(Zrk , ±{ rk/2 }). Let r = 8t + 7 and k = 4s + 3. The connection sets are given by Ωi = ±{1, 2, r + 1, r + 2} ⊕ 2ir, Ωi,j = ±{3, 4, 5, 6} ⊕ 4i ⊕ rj,. i = 0, . . . , s; i = 0, . . . , 2t and j = 0, . . . , 2s;. Ωi = ±{(2s + 1)r + 3, (2s + 1)r + 4, (2s + 1)r + 5, (2s + 1)r + 6} ⊕ 4i, i = 0, . . . , t − 1.. . 5 Maximum cyclic 4-cycle packings of Kr(k) of even order By Theorem 3.1, it suffices to consider the cases when r is even and k is odd. This implies that the leave of a maximum cyclic 4-cycle packing of Kr(k) must include a 1-factor of Kr(k) since the degree of each vertex in Kr(k) is odd. It is clear that the 1-factor must be the circulant graph X(Zrk , ±{rk/2}). Lemma 5.1 (1) If r ≡ 4 (mod 8) and k ≡ 3 (mod 4) or r ≡ 2 (mod 4) (r > 2) and k ≡ 7 (mod 8), then the leave of a maximum cyclic 4-cycle packing of Kr(k) is the union of a 1-factor and the circulant graph X(Zrk , ±{a}) with a even. (2) If r ≡ 2 (mod 4) and k ≡ 3 (mod 8), then the leave of a maximum cyclic 4-cycle packing of Kr(k) is the union of a 1-factor and the circulant graph X(Zrk , ±{a}) with a odd. (3) If r ≡ 2 (mod 4) and k ≡ 5 (mod 8), then the leave of a maximum cyclic 4-cycle packing of Kr(k) is the union of a 1-factor and the circulant graph X(Zrk , ±{a, b}) with a, b odd. Proof We consider only the case when r ≡ 4 (mod 8) and k ≡ 3 (mod 4) and leave the remainder to the reader. An easy computation shows that the numbers of odd and even differences in Kr(k) \X(Zrk , ±{rk/2}) are both odd, say α and β, and α − β ≡ 2 (mod 4). Set α − β = 4p + 2, p ≥ 0. By virtue of Lemma 2.3, the weight of any 4cycle C is a divisor of 4, i.e., W (C) = 1, 2, or 4. Note that if W (C) = 2, then two distinct differences in C must have the same parity since its index rk/2 is even. In order to obtain a maximum cyclic 4-cycle packing of Kr(k) , it is necessary to use β − 1 odd differences and β − 1 even differences to construct 4-cycles having weight 4, and then construct p 4-cycles each having weight 4 and all odd differences. Next, consider the remaining graph, that is, the circulant graph X(Zrk , Ω = ±{a, b, c, d}), where exactly one of elements in Ω, say a, is even and the rest is all odd. The proof then follows from Lemmas 2.4 and 2.6 by constructing the circulant graphs X(Zrk , ±{b}) with b = rk/4 and X(Zrk , ±{c, d}) with c + d = rk/2. Remark that by Lemma 2.4, the circulant graph X(Zrk , ±{a}) with rk and a both even is not a Hamiltonian cycle. It is not difficult to see that if r = 2 and k ≡ 7 (mod 8), then the leave of a maximum cyclic 4-cycle packing of K2(k) is the union.

(10) 374. J Comb Optim (2007) 14: 365–382. of a 1-factor and the circular graph X(Zrk , ±{a, b, c}) with a, b, c odd. Moreover, the leave of a maximum cyclic 4-cycle packing of Kr(k) is a 1-factor whenever r ≡ 0 (mod 8) and k ≡ 1 (mod 2), r ≡ 2 (mod 4) and k ≡ 1 (mod 8), or r ≡ 4 (mod 8) and k ≡ 1 (mod 4). Since the technique of proofs is analogous, in what follows, we shall list the connection sets without the details of verification. Furthermore, since the consequences in Lemma 5.1 will be repeatedly used later, for simplicity, we will not mention these again. Proposition 5.2 If r ≡ 0 (mod 8) and k ≡ 1 (mod 2), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a 1-factor. Proof Let r = 8t. The proof is divided into 4 cases according to whether k ≡ 3, 5, 7, or 1 (mod 8). Case 1. k ≡ 3 (mod 8), say k = 8s + 3. Ωa,i = ±{1 + i, rk/2 − 1 − i}, Ωb,i = ±{4, 5, 6, 7} ⊕ 4i,. i = 0, 1, 2;. i = 0, . . . , 2t − 2;. Ωc,i = ±{(4s + 1)r + 1, (4s + 1)r + 2, (4s + 1)r + 3, (4s + 1)r + 4} ⊕ 4i, i = 0, . . . , t − 2; Ωd,i = ±{r + 1, 2r + 1, 3r + 1, 4r + 1} ⊕ 4ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, 2r + 2, 2r + 3} ⊕ 2i ⊕ 2j r, i = 0, . . . , 4t − 2 and j = 0, . . . , 2s − 1. Case 2. k ≡ 5 (mod 8), say k = 8s + 5. Ωa = ±{rk/4}; Ωb = ±{rk/4 − 1, rk/4 + 1}; Ωc,i = ±{(2s + 1)r + 1, (2s + 1)r + 2, rk/4 + 2, rk/4 + 3} ⊕ 2i, i = 0, . . . , t − 2; Ωd,i = ±{rk/4 + 2t, rk/4 + 2t + 1, rk/4 + 2t + 2, rk/4 + 2t + 3} ⊕ 4i, i = 0, . . . , t − 1; Ωe = ±{1, rk/2 − 1}; Ωf = ±{2, 3, (4s + 2)r + 1, (4s + 2)r + 2}; Ωg,i = ±{4, 5, 6, 7} ⊕ 4i,. i = 0, . . . , 2t − 2;. Ωh = ±{(4s + 2)r + 3, (4s + 2)r + 4, (4s + 2)r + 5, (4s + 2)r + 6} ⊕ 4i, i = 0, . . . , t − 2; Ωi = ±{r + 1, 2r + 1, (2s + 2)r + 1, (2s + 3)r + 1} ⊕ 2ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, (2s + 2)r + 2, (2s + 2)r + 3} ⊕ 2i ⊕ rj, i = 0, . . . , (r − 4)/2 and j = 0, . . . , 2s − 1..

(11) J Comb Optim (2007) 14: 365–382. 375. Case 3. k ≡ 7 (mod 8), say k = 8s + 7. Ωi = ±{1, r + 1, 2r + 1, 3r + 1} ⊕ 4ir, Ωi. = ±{2, 3, r + 2, r + 3} ⊕ 2ir,. Ωi,j = ±{4, 5, 6, 7} ⊕ 4i ⊕ rj,. i = 0, . . . , s;. i = 0, . . . , 2s + 1; i = 0, . . . , 2t − 2 and j = 0, . . . , 4s + 2;. Ωi = ±{(4s + 3)r + 4, (4s + 3)r + 5, (4s + 3)r + 6, (4s + 3)r + 7} ⊕ 4i, i = 0, . . . , t − 2. Case 4. k ≡ 1 (mod 8), say k = 8s + 9. Ωa = ±{rk/4}; Ωb = ±{rk/4 − 1, rk/4 + 1}; Ωc,i = ±{(2s + 2)r + 1, (2s + 2)r + 2, rk/4 + 2, rk/4 + 3} ⊕ 2i, i = 0, . . . , t − 2; Ωd,i = ±{rk/4 + 2t, rk/4 + 2t + 1, rk/4 + 2t + 2, rk/4 + 2t + 3} ⊕ 4i, i = 0, . . . , t − 1; Ωe,i = ±{1, r + 1, (2s + 3)r + 1, (2s + 4)r + 1} ⊕ 2ir,. i = 0, . . . , s;. Ωf = ±{(2s + 1)r + 2, (2s + 1)r + 3, 4(s + 1)r + 2, 4(s + 1)r + 3}; Ωg,i = ±{(2s + 1)r + 4, (2s + 1)r + 5, (2s + 1)r + 6, (2s + 1)r + 7} ⊕ 4i, i = 0, . . . , 2t − 2; Ωh = ±{4(s + 1)r + 4, 4(s + 1)r + 5, 4(s + 1)r + 6, 4(s + 1)r + 7} ⊕ 4i, i = 0, . . . , t − 2; Ωi,j = ±{2, 3, (2s + 3)r + 2, (2s + 3)r + 3} ⊕ 2i ⊕ rj, i = 0, . . . , 4t − 2 and j = 0, . . . , 2s.. . When r ≡ 2 (mod 8) and k ≡ 1, 3, 5, or 7 (mod 8), the proof will be split into two cases according to whether r = 2 or r > 2. Proposition 5.3 If r ≡ 2 (mod 8) and k ≡ 1 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a 1-factor. Proof Let k = 8s + 9. Case 1. r = 2. Ωi = ±{1, 3, 5, 7} ⊕ 8i,. i = 0, . . . , s.. Case 2. r > 2, say r = 8t + 10. Ωi = ±{1, r + 1, 2r + 1, 3r + 1} ⊕ 4ir, Ωi,j = ±{2, 3, 4, 5} ⊕ 4i ⊕ rj,. i = 0, . . . , s;. i = 0, . . . , 2t + 1 and j = 0, . . . , 4s + 3;.

(12) 376. J Comb Optim (2007) 14: 365–382. Ωi = ±{4(s + 1)r + 1, 4(s + 1)r + 2, 4(s + 1)r + 3, 4(s + 1)r + 4} ⊕ 4i, i = 0, . . . , t.. . Proposition 5.4 If r ≡ 2 (mod 8) and k ≡ 3 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and a Hamiltonian cycle. Proof The Hamiltonian cycle is the circulant graph X(Zrk , ±{1}). Let k = 8s + 3. Case 1. r = 2. Ωi = ±{3, 5, 7, 9} ⊕ 8i,. i = 0, . . . , s − 1.. Case 2. r > 2, say r = 8t + 10. Ωa,i = ±{2, 3, 4, 5} ⊕ 4i,. i = 0, . . . , 2t + 1;. Ωb,i = ±{r + 1, 2r + 1, 3r + 1, 4r + 1} ⊕ 4ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, r + 4, r + 5} ⊕ 4i ⊕ rj, i = 0, . . . , 2t + 1 and j = 0, . . . , 4s − 1; Ωc,i = ±{(4s + 1)r + 1, (4s + 1)r + 2, (4s + 1)r + 3, (4s + 1)r + 4} ⊕ 4i, i = 0, . . . , t.. . Proposition 5.5 If r ≡ 2 (mod 8) and k ≡ 5 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 2 Hamiltonian cycles. Proof The 2 Hamiltonian cycles are respectively the circulant graphs X(Zrk , ±{1}) and X(Zrk , ±{rk/2 − 2}). Let k = 8s + 5. Case 1. r = 2. Ωi = ±{3, 5, 7, 9} ⊕ 8i,. i = 0, . . . , s − 1.. Case 2. r > 2, say r = 8t + 10. Ωa,i = ±{2, 3, 4, 5} ⊕ 4i,. i = 0, . . . , 2t + 1;. Ωb,i = ±{r + 1, 2r + 1, 3r + 1, 4r + 1} ⊕ 4ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, r + 4, r + 5} ⊕ 4i ⊕ rj, i = 0, . . . , 2t + 1 and j = 0, . . . , 4s − 1; Ωc,i = ±{(4s + 1)r + 1, (4s + 1)r + 2, (4s + 1)r + 3, (4s + 1)r + 4} ⊕ 4i, i = 0, . . . , 2t + 1; Ω = ±{(4s + 2)r − 1, (4s + 2)r + 1, rk/2 − 3, rk/2 − 1}; Ωd,j = ±{(4s + 2)r + 2, (4s + 2)r + 3, (4s + 2)r + 4, (4s + 2)r + 5} ⊕ 4i, i = 0, . . . , t − 1.. .

(13) J Comb Optim (2007) 14: 365–382. 377. Proposition 5.6 (1) If r = 2 and k ≡ 7 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 3 Hamiltonian cycles. (2) If r ≡ 2 (mod 8) (r > 2) and k ≡ 7 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 2 (rk/2)-cycles. Proof Let k = 8s + 7. (1) The 3 Hamiltonian cycles are respectively the circulant graphs X(Zrk , ±{1}), X(Zrk , ±{4s + 3}), and X(Zrk , ±{8s + 5}). Ωi = ±{3, 5, 4s + 5, 4s + 7} ⊕ 4i,. i = 0, . . . , s − 1.. (2) By virtue of Lemma 2.4, the circulant graph X(Zrk , ±{2}) is the union of 2 (rk/2)-cycles. Set r = 8t + 10. Ωa = ±{3, rk/2 − 3}; Ωb = ±{4, 5, rk/2 − 2, rk/2 − 1}; Ωc,i = ±{6, 7, 8, 9} ⊕ 4i,. i = 0, . . . , 2t;. Ωd,i = ±{(4s + 3)r + 2, (4s + 3)r + 3, (4s + 3)r + 4, (4s + 3)r + 5} ⊕ 4i, i = 0, . . . , t − 1; Ωe,i = ±{1, r + 1, 2r + 1, 3r + 1} ⊕ 4ir,. i = 0, . . . , s;. Ωi,j = ±{r + 2, r + 3, r + 4, r + 5} ⊕ 4i ⊕ rj, i = 0, . . . , 2t + 1 and j = 0, . . . , 4s + 1.. . Proposition 5.7 If r ≡ 4 (mod 8) and k ≡ 1 or 5 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a 1-factor. Proof We break the proof into two cases according to whether k ≡ 1 or 5 (mod 8). Let r = 8t + 4. Case 1. k ≡ 1 (mod 8), say k = 8s + 9. Ω = ±{rk/4}; Ωa,i = ±{1, r + 1, (2s + 3)r + 1, (2s + 4)r + 1} ⊕ 2ir, Ωb,i = ±{2, 3, (2s + 2)r + 1, (2s + 2)r + 2} ⊕ 2i,. i = 0, . . . , s;. i = 0, . . . , t − 1;. Ωc,i = ±{2t + 2, 2t + 3, (2s + 2)r + 2t + 2, (2s + 2)r + 2t + 3} ⊕ 2i, i = 0, . . . , 3t; Ωi,j = ±{r + 2, r + 3, (2s + 3)r + 2, (2s + 3)r + 3} ⊕ 2i ⊕ rj, i = 0, . . . , 4t and j = 0, . . . , 2s; Ωd,i = ±{(4s + 4)r + 2, (4s + 4)r + 3, (4s + 4)r + 4, (4s + 4)r + 5} ⊕ 4i, i = 0, . . . , t − 1..

(14) 378. J Comb Optim (2007) 14: 365–382. Case 2. k ≡ 5 (mod 8), say k = 8s + 5. Ωa = ±{1, rk/2 − 1}; Ωb = ±{rk/4}; Ωc,i = ±{2, 3, (2s + 1)r + 1, (2s + 1)r + 2} ⊕ 2i,. i = 0, . . . , t − 1;. Ωd,i = ±{2t + 2, 2t + 3, (2s + 1)r + 2t + 2, (2s + 1)r + 2t + 3} ⊕ 2i, i = 0, . . . , 3t; Ωe,i = ±{r + 1, 2r + 1, (2s + 2)r + 1, (2s + 3)r + 1} ⊕ 2ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, (2s + 2)r + 2, (2s + 2)r + 3} ⊕ 2i ⊕ rj, i = 0, . . . , 4t and j = 0, . . . , 2s − 1; Ωf,i = ±{(4s + 2)r + 1, (4s + 2)r + 2, (4s + 2)r + 3, (4s + 2)r + 4} ⊕ 4i, i = 0, . . . , t − 1.. . Proposition 5.8 If r ≡ 4 (mod 8) and k ≡ 3 or 7 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 2 (rk/2)cycles. Proof The circulant graph X(Zrk , ±{2}) is the union of 2 (rk/2)-cycles. The proof is split into two cases depending on whether k ≡ 3 or 7 (mod 8). Let r = 8t + 4. Case 1. k ≡ 3 (mod 8). Subcase 1.1. k = 3. Ω = ±{1, 3r/2 − 1}; Ω = ±{3r/4}; Ωa,i = ±{3, 4, 3r/4 + 1, 3r/4 + 2} ⊕ 2i,. i = 0, . . . , t − 1;. Ωb,i = ±{2t + 3, 2t + 4, 2t + 5, 2t + 6} ⊕ 4i, Ωc,i = ±{r + 1, r + 2, r + 3, r + 4} ⊕ 4i,. i = 0, . . . , t − 1;. i = 0, . . . , t − 1.. Subcase 1.2. k > 3, say k = 8s + 3. Ω = ±{1, rk/2 − 1}; Ω = ±{rk/4}; Ωa,i = ±{r − 1, 2r − 1, (2s + 2)r − 1, (2s + 3)r − 1} ⊕ 2ir, Ωb,i = ±{3, 4, 5, 6} ⊕ 4i,. i = 0, . . . , s − 1;. i = 0, . . . , 2t − 1;. Ωc,i = ±{(4s + 1)r + 1, (4s + 1)r + 2, (4s + 1)r + 3, (4s + 1)r + 4} ⊕ 4i, i = 0, . . . , t − 1; Ωi,j = ±{r + 1, r + 2, (2s + 1)r + 1, (2s + 1)r + 2} ⊕ 2i ⊕ rj, i = 0, . . . , 4t and j = 0, . . . , 2s − 2;.

(15) J Comb Optim (2007) 14: 365–382. 379. Ωd,i = ±{2sr + 1, 2sr + 2, 4sr + 1, 4sr + 2} ⊕ 2i,. i = 0, . . . , 3t;. Ωe,i = ±{2sr + 6t + 4, 2sr + 6t + 5, 4sr + 6t + 3, 4sr + 6t + 4} ⊕ 2i, i = 0, . . . , t − 1. Case 2. k ≡ 7 (mod 8), say k = 8s + 7. Ωa = ±{1, 3, (4s + 3)r − 1, (4s + 3)r + 1}; Ωb = ±{rk/4}; Ωc,i = ±{4, 5, 6, 7} ⊕ 4i,. i = 0, . . . , 2t − 1;. Ωd,i = ±{(4s + 3)r + 2, (4s + 3)r + 3, (4s + 3)r + 4, (4s + 3)r + 5} ⊕ 4i, i = 0, . . . , t − 1; Ωe,i = ±{r + 1, 2r + 1, (2s + 2)r + 1, (2s + 3)r + 1} ⊕ 2ir,. i = 0, . . . , s − 1;. Ωi,j = ±{r + 2, r + 3, (2s + 2)r + 2, (2s + 2)r + 3} ⊕ 2i ⊕ rj, i = 0, . . . , 4t and j = 0, . . . , 2s − 1; Ωf,i = ±{(2s + 1)r + 1, (2s + 1)r + 2, (4s + 2)r + 1, (4s + 2)r + 2} ⊕ 2i, i = 0, . . . , 3t; Ωg,i = ±{rk/4 + 1, rk/4 + 2, (4s + 2)r + 6t + 3, (4s + 2)r + 6t + 4} ⊕ 2i, i = 0, . . . , t − 1.. . Proposition 5.9 If r ≡ 6 (mod 8) and k ≡ 1 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave a 1-factor. Proof Let r = 8t + 6 and k = 8s + 9. Ω = ±{(rk − 2)/4, (rk + 2)/4}; Ωa,i = ±{r − 1, 2r − 1, 3r − 1, 4r − 1} ⊕ 4ir, Ωi,j = ±{1, 2, 3, 4} ⊕ 4i ⊕ rj, Ωi,j. i = 0, . . . , s;. i = 0, . . . , 2t and j = 0, . . . , 2s + 1;. = ±{(2s + 3)r + 1, (2s + 3)r + 2, (2s + 3)r + 3, (2s + 3)r + 4} ⊕ 4i ⊕ rj,. i = 0, . . . , 2t and j = 0, . . . , 2s; Ωb,i = ±{(2s + 2)r + 1, (2s + 2)r + 2, (rk + 2)/4 + 1, (rk + 2)/4 + 2} ⊕ 2i, i = 0, . . . , t − 1; Ωc,i = ±{(rk + 2)/4 + 2t + 1, (rk + 2)/4 + 2t + 2, (4s + 4)r + 1, (4s + 4)r + 2} ⊕ 2i,. i = 0, . . . , 2t.. . Proposition 5.10 If r ≡ 6 (mod 8) and k ≡ 3 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and a Hamiltonian cycle..

(16) 380. J Comb Optim (2007) 14: 365–382. Proof The Hamiltonian cycle is the circulant graph X(Zrk , ±{1}). Let r = 8t + 6 and k = 8s + 3. Ω = ±{(rk − 2)/4, (rk + 2)/4}; Ωa,i = ±{r + 1, 2r + 1, (2s + 1)r + 1, (2s + 2)r + 1} ⊕ 2ir, Ωi,j = ±{2, 3, (2s + 1)r + 2, (2s + 1)r + 3} ⊕ 2i ⊕ rj,. i = 0, . . . , s − 1;. i = 0, . . . , 4t + 1 and. j = 0, . . . , 2s − 1; Ωb,i = ±{2sr + 2, 2sr + 3, (rk + 2)/4 + 1, (rk + 2)/4 + 2} ⊕ 2i, i = 0, . . . , t − 1; Ωc,i = ±{2sr + 2t + 2, 2sr + 2t + 3, (4s + 1)r + 1, (4s + 1)r + 2} ⊕ 2i, i = 0, . . . , 2t.. . Proposition 5.11 If r ≡ 6 (mod 8) and k ≡ 5 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 2 Hamiltonian cycles. Proof The Hamiltonian cycles are the circulant graphs X(Zrk , ±{rk/2 − 2}) and X(Zrk , ±{(rk − 2)/4}). Let r = 8t + 6 and k = 8s + 5. Ω = ±{1, rk/2 − 1}; Ωa,i = ±{r + 1, 2r + 1, (2s + 2)r + 1, (2s + 3)r + 1} ⊕ 2ir, Ωi,j = ±{2, 3, 4, 5} ⊕ 4i ⊕ rj,. i = 0, . . . , s − 1;. i = 0, . . . , 2t and j = 0, . . . , 2s;. = ±{(2s + 2)r + 2, (2s + 2)r + 3, (2s + 2)r + 4, (2s + 2)r + 5} ⊕ 4i ⊕ rj, Ωi,j. i = 0, . . . , 2t and j = 0, . . . , 2s − 1; Ωb,i = ±{(2s + 1)r + 1, (2s + 1)r + 2, (rk − 2)/4 + 1, (rk − 2)/4 + 2} ⊕ 2i, i = 0, . . . , t − 1; Ωc,i = ±{(rk − 2)/4 + 2t + 1, (rk − 2)/4 + 2t + 2, (4s + 2)r + 1, (4s + 2)r + 2} ⊕ 2i,. i = 0, . . . , 2t − 1;. Ωd = ±{(rk − 2)/4 + 6t + 1, (rk − 2)/4 + 6t + 2, (rk − 2)/4 + 6t + 3, (rk − 2)/4 + 6t + 4}.. . Proposition 5.12 If r ≡ 6 (mod 8) and k ≡ 7 (mod 8), then there exists a maximum cyclic 4-cycle packing of Kr(k) with leave the union of a 1-factor and 2 (rk/2)-cycles. Proof The circulant graph X(Zrk , ±{rk/2 − 1}) is the union of 2 (rk/2)-cycles. Let r = 8t + 6 and k = 8s + 7. Ωa,i = ±{1, r + 1, 2r + 1, 3r + 1} ⊕ 4ir, Ωi,j = ±{2, 3, 4, 5} ⊕ 4i ⊕ rj,. i = 0, . . . , s;. i = 0, . . . , 2t and j = 0, . . . , 4s + 2;.

(17) J Comb Optim (2007) 14: 365–382. 381. Ωb,i = ±{(4s + 3)r + 2, (4s + 3)r + 3, (4s + 3)r + 4, (4s + 3)r + 5} ⊕ 4i, i = 0, . . . , t − 1.. . 6 Conclusion Combining Lemma 2.1, Theorem 3.1, and Propositions 4.2 to 4.6 and 5.2 to 5.12, we have the following main result. Theorem 6.1 There exists a maximum cyclic 4-cycle packing of the balanced complete multipartite graph Kr(k) with leave L where L is obtained as follows: (1) L is the empty set if k is even or k is odd and r ≡ 1 (mod 8); (2) L is 3 Hamiltonian cycles if r ≡ 3 (mod 8) and k ≡ 3 (mod 4) or r ≡ 7 (mod 8) and k ≡ 1 (mod 4); (3) L is 2 Hamiltonian cycles if r ≡ 5 (mod 8) and k ≡ 1 (mod 2); (4) L is a Hamiltonian cycle if r ≡ 3 (mod 8) and k ≡ 1 (mod 4) or r ≡ 7 (mod 8) and k ≡ 3 (mod 4); (5) L is a 1-factor if r ≡ 0 (mod 8) and k ≡ 1 (mod 2), r ≡ 2 (mod 4) and k ≡ 1 (mod 8), or r ≡ 4 (mod 8) and k ≡ 1 (mod 4); (6) L is the union of a 1-factor and 3 Hamiltonian cycles if r = 2 and k ≡ 7 (mod 8); (7) L is the union of a 1-factor and 2 Hamiltonian cycles if r ≡ 2 (mod 4) and k ≡ 5 (mod 8); (8) L is the union of a 1-factor and a Hamiltonian cycle if r ≡ 2 (mod 4) and k ≡ 3 (mod 8); and (9) L is the union of a 1-factor and 2 (rk/2)-cycles if r ≡ 2 (mod 4) (>2) and k ≡ 7 (mod 8) or r ≡ 4 (mod 8) and k ≡ 3 (mod 4).. References Alspach B, Gavlas H (2001) Cycle decompositions of Kn and Kn – I. J Comb Theory Ser B 81:77–99 Billington EJ (1999) Decomposing complete tripartite graphs into cycles of length 3 and 4. Discret Math 197/198:123–135 Billington EJ, Lindner CC (1996) Maximum packing of uniform group divisible triple systems. J Comb Des 4:397–404 Billington EJ, Fu H-L, Rodger CA (2001) Packing complete multipartite graphs with 4-cycles. J Comb Des 9:107–127 Billington EJ, Fu H-L, Rodger CA (2005) Packing λ-fold complete multipartite graphs with 4-cycles. Graphs Comb 21:169–185 Bryant D, Gavlas H, Ling A (2003) Skolem-type difference sets for cycle systems. Electron J Comb 10:1– 12 Buratti M (2003) Rotational k-cycle systems of order v < 3h; another proof of the existence of odd cycle systems. J Comb Des 11:433–441 Buratti M (2004) Existence of 1-rotational k-cycle systems of the complete graph. Graphs Comb 20:41–46 Buratti M, Del Fra A (2003) Existence of cyclic k-cycle systems of the complete graph. Discret Math 261:113–125 Buratti M, Del Fra A (2004) Cyclic Hamiltonian cycle systems of the complete graph. Discret Math 279:107–119 Cavenagh NJ (1998) Decompositions of complete tripartite graphs into k-cycles. Australas J Comb 18:193–200.

(18) 382. J Comb Optim (2007) 14: 365–382. Cavenagh NJ (2002) Further decompositions of complete tripartite graphs into 5-cycles. Discret Math 256:55–81 Cavenagh NJ, Billington EJ (2000) Decompositions of complete multipartite graphs into cycles of even length. Graphs Comb 16:49–65 Cavenagh NJ, Billington EJ (2002) On decomposing complete tripartite graphs into 5-cycles. Australas J Comb 22:41–62 Colbourn CJ, Wan P-J (2001) Minimizing drop cost for SONET/WDM networks with 1/8 wavelength requirements. Networks 37:107–116 Fu H-L, Huang W-C (2004) Packing balanced complete multipartite graphs with hexagons. Ars Comb 71:49–64 Fu H-L, Wu S-L (2004) Cyclically decomposing the complete graph into cycles. Discret Math 282:267– 273 Hoffman DG, Linder CC, Rodger CA (1989) On the construction of odd cycle systems. J Graph Theory 13:417–426 Kotzig A (1965) Decompositions of a complete graph into 4k-gons. Mat Fyz Casopis Sloven Akad Vied 15:229–233 (in Russian) Laskar R (1978) Decomposition of some composite graphs into Hamiltonian cycles. In: Hajnal A, Sos VT (eds) Proceedings of the fifth Hungarian coll., Keszthely, 1976. North-Holland, Amsterdam, pp 705– 716 Mahmoodian ES, Mirzakhani M (1995) Decomposition of complete tripartite graph into 5-cycles. In: Combinatorics advances. Kluwer Academic, Netherlands, pp 235–241 Mutoh Y, Morihara T, Jimbo M, Fu H-L (2003) The existence of 2 × 4 grid-block designs and their applications. SIAM J Discret Math 16:173–178 Peltesohn R (1938) Eine Lösung der beiden Heffterschen Differenzenprobleme. Compos Math 6:251–257 Ree DH (1967) Some designs of use in serology. Biometrics 23:779–791 Rosa A (1966a) On cyclic decompositions of the complete graph into (4m + 2)-gons. Mat Fyz Casopis Sloven Akad Vied 16:349–352 Rosa A (1966b) On cyclic decompositions of the complete graph into polygons with odd number of edges. ˇ Casopis P˘est Mat 91:53–63 (in Slovak) Šajna M (2002) Cycle decompositions, III: complete graphs and fixed length cycles. J Comb Des 10:27–78 ∗ ) into cycles (circuits) of length 2k. J Comb Theory Ser Sotteau D (1981) Decomposition of Km,n (Km,n B 30:75–81 Vietri A (2004) Cyclic k-cycle system of order 2km + k; a solution of the last open cases. J Comb Des 12:299–310 Wan P-J (1999) Multichannel optical networks: network theory and application, Kluwer Academic, Dordrecht Wu S-L, Fu H-L (2006) Cyclic m-cycle systems with m ≤ 32 or m = 2q with q a prime power. J Comb Des 14:66–81.

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