DOI 10.1007/s10878-007-9049-5
Packing 5-cycles into balanced complete m-partite
graphs for odd m
Ming-Hway Huang· Chin-Mei Fu · Hung-Lin Fu
Published online: 31 March 2007
© Springer Science+Business Media, LLC 2007
Abstract Let Kn1,n2,...,nm be a complete m-partite graph with partite sets of sizes
n1, n2, . . . , nm. A complete m-partite graph is balanced if each partite set has n
ver-tices. We denote this complete m-partite graph by Km(n). In this paper, we completely
solve the problem of finding a maximum packing of the balanced complete m-partite graph Km(n), m odd, with edge-disjoint 5-cycles and we explicitly give the minimum
leaves.
Keywords Complete m-partite graph· Balanced complete m-partite graph ·
5-cycle· Packing · Leave · Decomposition
1 Introduction
A few definitions, although many of them are standard, are first given for clarity. Let
Kmbe a complete graph with m vertices. A graph G is bipartite if V (G) is the union of two disjoint sets such that each edge consists of one vertex from each set. Let
Kn1,n2,...,nm be a complete m-partite graph with partite sets of sizes n1, n2, . . . , nm. A complete m-partite graph is balanced if each partite set has n vertices. We denote
Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Research of M.-H.W. was supported by NSC 93-2115-M-264-001. M.-H. Huang
Department of Computer Science and Information Engineering, Yuanpei Institute of Science and Technology, Hsinchu, Taiwan
C.-M. Fu (
)Department of Mathematics, Tamkang University, Tamsui, Taipei Shien, Taiwan e-mail: cmfu@math.tku.edu.tw
H.-L. Fu
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan e-mail: cmfu@mail.tku.edu.tw
this complete m-partite graph by Km(n). A subgraph of graph G is a graph H such
that V (H )⊆ V (G) and E(H ) ⊆ E(G); an induced subgraph H of G is a subgraph of G such that E(H ) consists of all edges of G whose end points belong to V (H ). If S is a nonempty set of vertices of G, then the subgraph of G induced by S is the induced subgraph of G with vertex set S. This induced subgraph of G is denoted by G[S].
A Latin square of order n based on an n-element set is an n× n array in which each cell contains a single element from the set such that each element occurs exactly once in each row and each column. A Latin square A= [ai,j] of order n based on the set Zn= {0, . . . , n − 1} is called idempotent if ai,i= i for each i ∈ Zn.
A k-cycle is a cycle of length k. A k-cycle packing of a graph G is a set of edge-disjoint k-cycles in G. A k-cycle packing C of G is maximum if|C| ≥ |C| for all other k-cycle packings Cof G. The leave L of a packing C is the subgraph induced by the set of edges of G that do not occur in any k-cycle of the packing C. The leave
Lof a maximum packing is referred to as a minimum leave, a leave with minimum number of edges. A packing with empty leave is known as a k-cycle system of G. A k-cycle system of a complete graph Kvwith v vertices is referred to as a k-cycle
system of order v.
Clearly, if Kv can be decomposed into a k-cycle system then v is odd and k
di-videsv2
. To determine whether the above necessary condition is also sufficient is commonly referred to as the existence problem of k-cycle system.
The existence problem for k-cycle system of order v has been studied for more than 35 years. Recently, it has been completely solved by Alspach et al. see (Alspach and Gavlas2001; Alspach and Marshall1994; Wilson1974). But, the packing of Kv
with k-cycles is not that lucky, only partial results are obtained so far, see (Lindner and Rodger1992). Mainly, k∈ {3,4,5,6} has been considered.
If we turn to the k-cycle packing of a complete multipartite graph, then the prob-lem is getting more difficult. Even in the case k= 3, the existence problem is still unsolved; see (Lindner and Rodger1992). Recently, Billington, Fu and Rodger com-pletely solved the case k= 4, see (Billington et al.2001,2005). The cases other than
k= 4 remain unsettled.
In this paper, we consider a 5-cycle packing of a balanced complete m-partite graph Km(n) for odd m and we obtain a minimum leave of a maximum packing
of Km(n). The following two results obtained by Cavenagh and Billington, Rosa and
Znám respectively are essential.
Theorem 1.1 (Cavenagh and Billington 2000b) The complete tripartite graph
Km1,m2,m3(with m1≤ m2≤ m3) can be decomposed into 5-cycles only if m1, m2, m3
are either all odd or all even, 5 divides|E(Km1,m2,m3)| and m3≤ 4m1m2/(m1+m2).
These necessary conditions are sufficient in the case when two partite sets have equal size or in the case when m1and m2are divisible by 10.
Theorem 1.2 (Rosa and Znám1994) The minimum leaves of the maximum packings
of Kvwith 5-cycles are as follows in Table1. v is considered to be the number
mod-ulo 10. F is a 1-factor, Ci is a cycle of length i, Fi is a graph with v/2+ i edges and
Table 1 The minimum leaves
of the maximum packings of Kv
with 5-cycles
v 0 1 2 3 4 5 6 7 8 9
L F ∅ F C3 F4 ∅ F2 2C3 F4 2C3
We note here that in the cases v≡ 7 or 9 (mod 10) the leave 2C3represents two
C3with one vertex in common. It is also known as a bowtie.
2 The maximum 5-cycle packing of Km(n)
First, we consider a maximum 5-cycle packing of Kn,n,n. Before that we need to
solve some small cases:
Lemma 2.1 There is a 5-cycle packing of K3,3,3with leave C3∪ C4.
Proof Let Z3× Z3be the vertex set of K3,3,3. Then K3,3,3 can be packed with
5-cycles: ((0, j ), (2, 1+j), (0, 2+j), (1, j), (1, 2+j)), j = 1, 2, ((0, 0), (0, 1), (0, 2),
(1, 0), (1, 2)) and ((1, 0), (2, 1), (2, 0), (1, 1), (2, 2)) with leave C3∪ C4: ((2, 0),
(2, 2), (2, 1), (1, 2))∪ ((2, 1), (0, 0), (0, 2)).
Lemma 2.2 There is a 5-cycle packing of K4,4,4with leave C3.
Proof Let Z4× Z3 be the vertex set of K4,4,4. Then K4,4,4 can be packed with
5-cycles: ((i, j ), (2 + i, 1 + j), (i, 2 + j), (1 + i, j), (1 + i, 2 + j)), i = 0, 1,
j ∈ Z3, ((0, 0), (3, 2), (2, 1), (3, 0), (3, 1)), ((2, 0), (3, 2), (3, 0), (0, 2), (3, 1)) and
((3, 0), (2, 2), (3, 1), (3, 2), (0, 1)) with leave C3: ((0, 0), (0, 1), (0, 2)). Lemma 2.3 There is a 5-cycle packing of K6,6,6with leave C3.
Proof Let ({∞} ∪ Z5)× Z3be the vertex set of K6,6,6. Then K6,6,6can be packed
with 5-cycles: ((i, j ), (∞, 1 + j), (i, 2 + j), (1 + i, j), (1 + i, 2 + j)), i ∈ Z5,
j∈ Z3, and ((0, j ), (3, 1+j), (1, j), (4, 1+j), (2, 2+j)), ((4, j), (1, 1+j), (3, j),
(0, 1+ j), (2, 2 + j)), j ∈ Z3with leave C3: ((∞, 0), (∞, 1), (∞, 2)). Lemma 2.4 There is a 5-cycle packing of K7,7,7with leave C3∪ C4.
Proof Let Z7 × Z3 be the vertex set of K7,7,7. Since K7,7,7 can be
decom-posed into K5,5,5 and three copies of K5,2,2. Let Z5× Z3 be the vertex set of K5,5,5. Then K5,5,5 can be decomposed into following 5-cycles: ((i, j ), (2+ i,
1 + j), (i, 2 + j), (1 + i, j), (1 + i, 2 + j)), i ∈ Z5, j ∈ Z3. Since K5,2,2 can
not be decomposed into 5-cycles, we can pack K5,2,2 with 5-cycles with leave a K1,4 or 2K1,2. Therefore K7,7,7 can be packed with 5-cycles with leave 2K1,4∪
2K1,2: ((4, 1), (5, 0)), ((4, 1), (6, 0)), ((4, 1), (5, 2)), ((4, 1), (6, 2)), ((0, 2), (5, 0)), ((0, 2), (6, 0)), ((0, 2), (5, 1)), ((0, 2), (6, 1)), ((3, 0), (6, 1)), ((3, 0), (6, 2)), ((4, 0),
(5, 1)), ((4, 0), (5, 2)). In the above packing of K5,5,5, there is a 5-cycle C5: ((3, 1), (0, 2), (3, 0), (4, 1), (4, 0)). Then 2K1,4∪ 2K1,2∪ C5can be packed with two
5-cycles: ((3, 0), (6, 1), (0, 2), (5, 0), (4, 1)) and ((6, 0), (4, 1), (6, 2), (3, 0), (0, 2)) with leave C3∪ C4: ((4, 0), (5, 2), (4, 1))∪ ((4, 0), (5, 1), (0, 2), (3, 1)).
Lemma 2.5 There is a 5-cycle packing of K9,9,9with leave C3.
Proof Let ({∞} ∪ Z8)× Z3be the vertex set of K9,9,9. Then K9,9,9can be packed
with 5-cycles: ((i, j ), (∞, 1 + j), (i, 2 + j), (1 + i, j), (1 + i, 2 + j)), and ((i, j),
(4+ i, 1 + j), (1 + i, 2 + j), (4 + i, j), (6 + i, 2 + j)), i ∈ Z8, j∈ Z3with leave C3:
((∞, 0), (∞, 1), (∞, 2)).
Lemma 2.6 There is a 5-cycle packing of K11,11,11with leave C3.
Proof Let the vertex set of K11,11,11be ({∞} ∪ Z10)× Z3, and let Ai= {5i, 5i + 1,
5i + 2, 5i + 3, 5i + 4}, for each i ∈ Z2. Since K11,11,11 can be decomposed
into K6,6,6 with vertex set ({∞} ∪ A0)× Z3 and three copies of K5,5,6, we can
pack K6,6,6 with leave C3: ((∞, 0), (∞, 1), (∞, 2)) and K5,5,6 with leave 6K1,5: ((∞, i), (5, j)), ((∞, i), (6, j)), ((∞, i), (7, j)), ((∞, i), (8, j)), ((∞, i), (9, j)),
for j = i + 1, i + 2, i = 0, 1, 2. Using the same construction as in the proof of Lemma 2.4, we can get two C5 from K5,5,6: ((2, 1), (5, 0), (6, 2), (6, 0), (5, 2)), ((4, 0), (7, 2), (8, 1), (8, 2), (7, 1)). Then C3 ∪ 6K1,5 ∪ 2C5 can be packed with
8C5: ((∞, 0), (∞, 1), (9, 0), (∞, 2), (7, 1)), ((∞, 1), (9, 2), (∞, 0), (9, 1), (∞, 2)), ((∞, 2), (∞, 0), (5, 2), (∞, 1), (8, 0)), ((∞, 0), (7, 2), (8, 1), (∞, 2), (6, 1)), ((4, 0), (7, 2), (∞, 1), (8, 2), (7, 1)), ((∞, 0), (6, 2), (6, 0), (∞, 2), (5, 1)), ((∞, 1), (6, 2),
(5, 0), (∞, 2), (7, 0)), ((∞, 1), (6, 0), (5, 2), (2, 1), (5, 0)) with leave C3: ((∞, 0),
(8, 1), (8, 2)).
Lemma 2.7 There is a 5-cycle packing of Kn,n,nwith leave (i) C3when n≡ 1 or 4 (mod 5) and (ii) C3∪ C4when n≡ 2 or 3 (mod 5).
Proof (i) n= 5k + 1, k ≥ 1. Let ({∞} ∪ Z5k)× Z3be the vertex set of Kn,n,n, and Ai = {5i, 5i + 1, 5i + 2, 5i + 3, 5i + 4}, for each i ∈ Zk. When k= 1 and k = 2, it
can be seen in Lemmas2.3and2.6respectively. If k≥ 3, let M = [mi,j] be an idem-potent Latin square of order k based on Zk. For each i∈ Zk, the induced subgraph Kn,n,n[({∞} ∪ Ai)× Z3] is isomorphic to K6,6,6. By Lemma2.3, Kn,n,n[({∞} ∪
Ai)× Z3] can be packed with 5-cycles with leave C3: ((∞, 0), (∞, 1), (∞, 2)) for each i. By Theorem1.1, the induced subgraph Kn,n,n[(Ai × {0}) ∪ (Aj× {1}) ∪
(Ami,j× {2})], which is isomorphic to K5,5,5, can be decomposed into 5-cycles for each i = j. This implies that Kn,n,ncan be packed with 5-cycles with leave C3.
(ii) n= 5k + 2, k ≥ 0. If n = 2, it is easy to see that K2,2,2can be packed with one 5-cycle which has leave C3∪C4. If k≥ 1, let the vertex set of Kn,n,nbe ({∞1,∞2}∪ Z5k)× Z3. Let A0= {∞1,∞2} ∪ {0, 1, 2, 3, 4} and Ai = {5i, 5i + 1, 5i + 2, 5i +
3, 5i+ 4}, for each i ∈ Zkand i≥ 1. Then the induced subgraph Kn,n,n[(Ai× {0}) ∪
(Aj× {1}) ∪ (Ai+j× {2})] is isomorphic to K7,7,7if i= 0 and j = 0, and isomorphic
to K5,5,5or K5,5,7otherwise. Therefore by Theorem1.1and Lemma2.4, we obtain
that Kn,n,ncan be packed with 5-cycles with leave C3∪ C4.
(iii) n= 5k +3, k ≥ 0. Let ({∞1,∞2,∞3}∪Z5k)×Z3be the vertex set of Kn,n,n.
Let A0= {∞1,∞2,∞3} and Ai+1= {5i, 5i + 1, 5i + 2, 5i + 3, 5i + 4}, for each i∈ Zk. Then the induced subgraph Kn,n,n[(Ai× {0}) ∪ (Aj× {1}) ∪ (Ai+j× {2})] is isomorphic to K3,3,3if i= 0 and j = 0, and isomorphic to K5,5,5or K5,5,3otherwise.
By Theorem1.1and Lemma2.1, we obtain that Kn,n,ncan be packed with 5-cycles
with leave C3∪ C4.
(iv) n= 5k + 4, k ≥ 0. Let the vertex set of Kn,n,n be ({∞1,∞2,∞3,∞4} ∪
Z5k)× Z3. Let A0= {∞1,∞2,∞3,∞4} ∪ {0, 1, 2, 3, 4} and Ai= {5i, 5i + 1, 5i +
2, 5i+ 3, 5i + 4}, for each i ∈ Zkand i≥ 1. Then the induced subgraph Kn,n,n[(Ai×
{0}) ∪ (Aj × {1}) ∪ (Ai+j× {2})] is isomorphic to K9,9,9if i= 0 and j = 0, and isomorphic to K5,5,5or K5,5,9otherwise. By Theorem1.1and Lemma2.5, we obtain
that Kn,n,ncan be packed with 5-cycles with leave C3.
Since the number of edges in the leave of above 5-cycle packing of Kn,n,nis the
minimum, we have finished the maximum 5-cycle packing of the balance complete tripartite graphs. Now, we go on to consider the following special graphs.
Lemma 2.8 Let n≥ 2, and C5(n)denote the graph with vertex set Zn× Z5and edge
set E(C5(n)), where{(i1, j1), (i2, j2)} ∈ E(C5(n))if and only if j2≡ j1+ 1 (mod 5).
Then C5(n)can be decomposed into 5-cycles.
Proof C5(n) can be decomposed into n2 5-cycles: {((i, 0), (j, 1), (i, 2), (j, 3),
(i+ j, 4)) | i, j, i + j ∈ Zn}.
Lemma2.8 gives us a good idea to pack a balanced complete m-partite graph
Km(n)with 5-cycles. If we view each partite set of Km(n)as a point, then it will turn
to be a complete graph Km of order m. By Theorem1.2, we can pack the complete graph Km with 5-cycles which has leave an Lm. Thus the leave of the packing of
Km(n) with 5-cycles depends on the leave of the packing of Lm(n) with 5-cycles.
Since the leave of the packing contains the fewest number of edges, we will get the maximum packing. Now we are ready for the maximum packing of Km(n)with
5-cycles, where m is odd.
Theorem 2.9 Let m be an odd integer. Then the minimum leaves of the maximum
packings of Km(n)with 5-cycles are as follows in Table2. m is considered to be the
number modulo 10, n is considered to be the number modulo 5.
Proof If we consider each partite set of Km(n)as a vertex, then Km(n)can be viewed
as the complete graph Km.
(1) m≡ 1 or 5 (mod 10). By Theorem1.2, Km can be decomposed into 5-cycles. By Lemma2.8, Km(n)can be decomposed into 5-cycles.
Table 2 The minimum leaves
of the maximum packings of Km(n)with 5-cycle m n 0 1 2 3 4 1 ∅ ∅ ∅ ∅ ∅ 3 ∅ C3 C3∪ C4 C3∪ C4 C3 5 ∅ ∅ ∅ ∅ ∅ 7 ∅ 2C3 C4 C4 2C3 9 ∅ 2C3 C4 C4 2C3
Fig. 1 Pack 2(C3∪ C4)with
two 5-cycles with leave a C4
Since Km(n)is a simple graph and m is odd, the degree of each vertex in Km(n)is
even. A nonempty leave of a 5-cycle packing of Km(n)contains at least 3 edges.
(2) m≡ 3 (mod 10). Then 5|(|E(Km )| − 3). By Theorem1.2, Km can be packed with 5-cycles with leave C3. Therefore, Km(n) can be packed with 5-cycles with
leave C3(n). C3(n)is isomorphic to Kn,n,n. By Theorem1.1and Lemma2.7, if n≡ 0
(mod 5), then Kn,n,ncan be decomposed into 5-cycles. Thus Km(n)can be
decom-posed into 5-cycles. If n≡ 1 or 4 (mod 5), then Kn,n,ncan be packed with 5-cycles with leave C3. Thus Km(n)can be packed with 5-cycles with leave C3. If n≡ 2 or 3
(mod 5), then Kn,n,n can be packed with 5-cycles with leave C3∪ C4. Thus Km(n)
can be packed with 5-cycles with leave C3∪ C4.
(3) m≡ 7 or 9 (mod 10). Then 5|(|E(Km )| − 6). By Theorem 1.2, Km can be packed with 5-cycles with leave 2C3. As noted earlier 2C3 is the union of two C3
with one vertex in common. By Theorem1.1and Lemma2.7, if n≡ 0 (mod 5), then
Km(n)can be decomposed into 5-cycles. If n≡ 1 or 4 (mod 5), then Km(n) can be
packed with 5-cycles with leave two 3-cycles. If n≡ 2 or 3 (mod 5), then Kn,n,ncan be packed with 5-cycles with leave C3∪ C4. From Fig.1, the two C3∪ C4 can be
decomposed into two 5-cycles and one C4. Thus Km(n)can be packed with 5-cycles
with leave a C4.
3 Concluding remark
A k-cycle covering of G is a triple (V (G),C, P ), where P ⊆ E(G) is called the padding, andC is a collection of k-cycles that partition E(G) + P . If |P | is the min-imum, then (V (G),C, P ) is called a minimum covering of G with k-cycles. There-fore, a k-cycle system of G is a k-cycle covering of G with padding P = ∅. A bit of reflection, if we have a k-cycle packing of G with leave L, then P is a padding provided that L∪ P can be decomposed into k-cycles. Therefore, by the result ob-tained in this paper we can also find a minimum 5-cycle covering of Km(n)for odd m
without too much difficulty.
As to the 5-cycle packing of Km(n) when m is even, due to the complexity of
leaves, it is much more complicated. We wish the problem can be solved in the near future.
Acknowledgement The authors wish to extend their gratitude to the referees for their helpful comments in revising this paper.
References
Alspach B, Gavlas H (2001) Cycle decompositions of Knand Kn− I . J Comb Theory Ser B 81:77–99
Alspach B, Marshall S (1994) Even cycle decompositions of complete graphs minus a 1-factor. J Comb Des 2:441–458
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
Cavenagh NJ, Billington EJ (2000b) On decomposing complete tripartite graphs into 5-cycles. Australas J Comb 22:41–62
Lindner CC, Rodger CA (1992) Decomposition into cycle II: cycle systems. In: Dinitz JH, Stinson DR (eds) Contemporary design theory: a collection of surveys. Wiley, New York, pp 325–369
Rosa A, Znám S (1994) Packing pentagons into complete graphs: how clumsy can you get. Discrete Math 128:305–316
Wilson RM (1974) Some partitions of all triples into Steiner triple systems. In: Lecture notes in mathe-matics, vol 411. Springer, Berlin, pp 267–277
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