Cyclic m-Cycle Systems of K n, n for

m 100

Yuge Zheng

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240,

P.R.China

Department of Mathematics, Henan Polytechnic University Jiaozuo 454000, P.R.China

1 Introduction An m-cycle system of a graph G is a set of m-

cycles in G whose edges partition the edge set of G. Let

K n;k denote the complete bipartite graph with partite

sets of sizes n and k. The existence problem for m-cycle

systems of K n;k were completely settled in [7](1981).

Theorem 1.1([7]) There exists an m-cycle system of K

n;k if and only if m; n and k are even,n; k m/2 and m

divides nk.

An m-cycle system of a graph G with vertex Zv is

cyclic if for each B = (b1; b2; …; bm) we have B +j

, where for each j Zv the sum B + j is defined as B + j

= (b1 + j; b2 + j; …; b m + j). In the sequel, any graph of

order v will be considered as a graph with vertex set Zv.

The existence problem for cyclic m-cycle systems of

complete graph has attracted much in-terest. For m

even and v 1 (mod 2m), cyclic m-cycle systems of Kv

were constructed for m 0 (mod 4) in [5] and for m 2

(mod 4) in [6]. For m odd and v 1 (mod 2m), cyclic m-

cycle systems of Kv were found using different methods

in [3,1,4]. For v m (mod 2m), cyclic m-cycle systems

of Kv were given in [2] for m M, where M = pe I p is

prime, e > 1 in [10] for m M.

For the existence of cyclic m-cycle systems of

complete bipartite graph K n;n necessary and suffcient

conditions were given in [8] for the case m 0

(mod4)

15

，

and m/ 4 square-free and the case m 2 (mod 4) with m >

= 6 and m square-free. In [8,9], necessary and suffcient

conditions are determined for the existence of cyclic m-

cycle systems of K n;n for all integers m 30.

In this paper, we will determine necessary and suffcient

conditions for the existence of cyclic m-cycle systems of K

n;n for all integers 30 < m 100. As a consequence,necessary

and suffcient conditions are determined for the existence of

cyclic m-cycle systems of K n;n for all integers m 100.

2 Preliminaries

The main technique used in this paper is the

difference method.

Definition2.1 The type of a cycle B is the cardinality

of the set { j Z∈ v lB=B+j }.

If B = (b1; b2; …;bm) is a m-cycle of type d,

let denote the multiset

where b0 = bm.

Definition 2.2 Let be a set of m-cycles

and di be the type of Bi for If each element in

Zv\ 0 appears exactly once in the multiset ,

then is called a (Kv;Cm)-

1{ ( ) 1,2, / }i ib b i m d

1,2, ,i l

-difference system((Kv;Cm)-DS for short).

If there exists a cyclic m-cycle systems of K n;n, then the two partite sets of K n;n be

Definition 2.3 Let be a set of m-cycles and di be the type of Bi for If each element

in appears exactly once in the multiset

then is called a (K n;n;Cm)-DS.

Proposition 2.4 If is a (K n;n;Cm)-DS, then the cycles form a cyclic

m-cycle system of Kn;n where di is the type of Bi.

0,2,4, , 2 2 1,3,5 ,2 1 .n and n

1,2, , ; 1, 2, , 2 /i iB j i l j n d

1,2, ,i l

　　

Theorem 2.5([9]) There exists a cyclic m-cycle system of Kn;n if and only if there exists a (Kn;n;Cm)-DS.

Theorem 2.6([8]) Let m, n be positive integers, m 0 (mod 4). There exists a cyclic m-cycle system of Kn;n for n

0 (mod m/2 ) .

Theorem 2.7([8]) Let m, n be positive integers, m 2 (mod 4) with m 6. There exists a cyclic m-cycle system of Kn;n for n 0(mod 2m).

Theorem 2.8([8]) Let m, n be positive integers, m 0 (mod 4) and m /4 is square-free. There exists a cyclic m-cycle system of Kn;n if and only if n 0 (mod m/2 ).

．

Theorem 2.9([8]) Let m, n be positive integers, m 2 (mod 4) with m >6 and m is square-free. There exists a cyclic m-cycle system of Kn;n if and only if n 0 (mod 2m).

Theorem 2.10([9]) Let m, n be positive integers, m 2 (mod 4). There is no cyclic m-cycle system of Kn;n for n 2 (mod 4).

Theorem 2.11([8]) There exists a cyclic 16-cycle systems of Kn;n if and only if n 0 (mod 8).

Theorem 2.12([9]) There exists a cyclic 18-cycle systems of Kn;n if and only if n 0 (mod 12)

3 Cyclic m-Cycle System of K n;n for m 0 (mod 4) and

Theorem 3.1 For m = 40; 44; 52; 56; 60; 68; 76; 84; 88; 92, there exists a cyclic m-cycle systems of Kn;n if and only if n 0 (mod m/2 ).

Lemma 3.2 There exists a cyclic 32-cycle systems of Kn;n for n 8 (mod 16), and n > 8.

Theorem 3.3 There exists a cyclic 32-cycle systems of Kn;n if and only if n 0 (mod 8) and n 16

Lemma 3.4 There exists a cyclic 36-cycle systems of Kn;n for n 6; 30 (mod 36) and n 30 .

30 100m

Lemma 3.5 There exists a cyclic 36-cycle systems of Kn;n for n 12; 24 (mod 36) and n 24.

Theorem 3.6 There exists a cyclic 36-cycle systems of Kn;n if and only if n 0 (mod 6) and n 18 .

Lemma 3.7 There is no cyclic 48-cycle system of Kn;n for n 12 (mod 48).

Lemma 3.8 There is no cyclic 48-cycle system of Kn;n for n 36 (mod 48).

Theorem 3.9 There exists a cyclic 48-cycle systems of Kn;n if and only if n 0(mod 24).

Lemma 3.10 There exists a cyclic 64-cycle systems of Kn;n for n 16 (mod 32) and n 48. Lemma 3.11 There is no cyclic 64-cycle system of Kn;n for n 8 (mod 32).

Lemma 3.12 There is no cyclic 64-cycle system of Kn;n for n 24 (mod 32).

Theorem 3.13 There exists a cyclic 64-cycle systems of Kn;n if and only if n 0(mod 16) and n 32.

Lemma 3.14 There exists a cyclic 72-cycle systems of Kn;n for n 0 (mod 12) and n 36

Theorem 3.15 There exists a cyclic 72-cycle systems of Kn;n if and only if n 0 (mod 12) and n 36.

Lemma 3.16 There is no cyclic 80-cycle system of Kn;n for n 20 (mod 40).

Theorem 3.17 There exists a cyclic 80-cycle systems of Kn;n if and only if n 0(mod 40)

Lemma 3.18 There exists a cyclic 96-cycle systems of Kn;n for n 24 (mod 48) and n 72.

Theorem 3.19 There exists a cyclic 96-cycle systems of Kn;n if and only if n 0 (mod 24) and n 48.

Lemma 3.20 There exists a cyclic 100-cycle systems of Kn;n for n 0 (mod 20) and n 60.

Lemma 3.21 There exists a cyclic 100-cycle systems of Kn;n for n 10 (mod 20) and n 70.

Theorem 3.22 There exists a cyclic 100-cycle systems of K n;n if and only if n 0 (mod 10) and n 50.

4 Cyclic m-Cycle System of K n;n for m 2 (mod 4) and 30 ＜ m 100

Theorem 4.1 For m = 34; 38; 42; 46; 58; 62; 66; 70; 74; 78; 82; 86; 94, there exists a cyclic m-cycle systems of Kn;n if and only if n 0 (mod 2m).

Lemma 4.2 If there exists a cyclic 50-cycle system of Kn;n, then n 0 (mod 20).

Lemma 4.3 There exists a cyclic 50-cycle systems of Kn;n for n 20; 60 (mod 100) and n 40.

Lemma 4.4 There exists a cyclic 50-cycle systems of Kn;n for n 40; 80 (mod 100).

Theorem 4.5 There exists a cyclic 50-cycle systems of Kn;n if and only if n 0(mod 20) and n 40 .

Lemma 4.6 If there exists a cyclic 54-cycle system of Kn;n, then n 0 (mod 36).

Lemma 4.7There exists a cyclic 54-cycle systems of Kn;n for n 36; 72 (mod 108).

Theorem 4.8 There exists a cyclic 54-cycle systems of Kn;n if and only if n 0(mod 36).

Lemma 4.9 If there exists a cyclic 90-cycle system of Kn;n, then n 0 (mod 60).

Lemma 4.10 There exists a cyclic 90-cycle systems of Kn;n for n 0 (mod 60) Theorem 4.11 There exists a cyclic 90-cycle systems of Kn;n if and only if n 0(mod 60).

Lemma 4.12 If there exists a cyclic 98-cycle system of Kn;n, then n 0; 28; 56; 84; 112; 140; (mod 196).

Lemma 4.13 There exists a cyclic 98-cycle systems of Kn;n for n 0 (mod 28) and n 56

Theorem 4.15 There exists a cyclic 98-cycle systems of K n;n if and only if n 0(mod 28) and n 56.

References

[1] A. Blinco, S. El Zanati, C. Vanden Eynden, On the cyclic decomposition of complete graphs into almost-bipartite graphs, Discrete Math. 284 (2004), 71-81.

[2] M. Buratti, A. DelFra, Cyclic hamiltonian cycle systems of the complete graph, Discrete Math. 279 (2004), 107-119.

[3] M. Buratti, A. DelFra, Existence of cyclic k-cycle systems of the complete graph, Discrete Math. 26 (2003), 113-225.

[4] H. Fu and S. Wu, Cyclically decomposing the complete graphs into cycles, Discrete Math. 282 (2004), 267-273.

[5] A. Kotzig, On decompositions of the complete graph into 4k-gons, Mat.-Fyz. Cas. 15 (1965),227-233

[6] A.Rosa, On cyclic decompositions of the complete graph into (4m + 2)-gons, Mat.-Fyz. Cas.16 (1966), 349-352.

[7] D. Sotteau, Decomposition of Km;n(K¤m;m into cycles(circuits) of length 2k, J. Combin. The-ory Ser. B 29 (1981),75-81.

[8] Wenwen Sun, Hao shen, Cyclic Cycle Systems of the Complete Bipartite Graph Kn;n, (preprint).

[9] Wenwen Sun, Cyclic m-Cycle Systems of Kn;n for m · 30 (preprint).

[10] A. Vietri, Cyclic k-cycle systems of order 2kn + k: a solution of the last open cases, J. Combin. Designs 12 (2004), 299-310.

This paper was done while the author was visiting

the Department of Mathematics, Shanghai Jiao Tong

University as a senior visiting scholar. The

author would like to thank Prof . Hao Shen

for his constructive suggestions.I would also like to

thank Dr. Xiuli Li and Dr . Wenwen Sun fortheir

help during the preparation of the paper.

Acknowledgement

Acknowledgement

谢 谢 ！