Cyclic m-Cycle Systems of K n, n for m 100 Yuge Zheng Department of Mathematics, Shanghai Jiao Tong University, Shanghai , P.R.China Department of Mathematics, Henan Polytechnic University Jiaozuo , 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 Z v is cyclic if for each B = (b 1 ; b 2 ; …; b m ) we have B +j, where for each j Z v the sum B + j is defined as B + j = (b 1 + j; b 2 + j; …; b m + j). In the sequel, any graph of order v will be considered as a graph with vertex set Z v.

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 K v 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 K v were found using different methods in [3,1,4]. For v m (mod 2m), cyclic m-cycle systems of K v were given in [2] for m M, where M = p e 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 (mod 4)

， 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 = (b 1 ; b 2 ; …;b m ) is a m-cycle of type d, let denote the multiset where b 0 = b m. Definition 2.2 Let be a set of m-cycles and di be the type of Bi for If each element in Z v \ 0 appears exactly once in the multiset, then is called a (K v ;C m )-

-difference system((K v ;C m )-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 ;C m )-DS. Proposition 2.4 If is a (K n;n ;C m )- DS, then the cycles form a cyclic m-cycle system of K n;n where di is the type of Bi.

Theorem 2.5([9]) There exists a cyclic m-cycle system of K n;n if and only if there exists a (K n;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 K n;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 K n;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 K n;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 K n;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 K n;n for n 2 (mod 4). Theorem 2.11([8]) There exists a cyclic 16-cycle systems of K n;n if and only if n 0 (mod 8). Theorem 2.12([9]) There exists a cyclic 18-cycle systems of K n;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 K n;n if and only if n 0 (mod m/2 ). Lemma 3.2 There exists a cyclic 32-cycle systems of K n;n for n 8 (mod 16), and n > 8. Theorem 3.3 There exists a cyclic 32-cycle systems of K n;n if and only if n 0 (mod 8) and n 16 Lemma 3.4 There exists a cyclic 36-cycle systems of K n;n for n 6; 30 (mod 36) and n 30.

Lemma 3.5 There exists a cyclic 36-cycle systems of K n;n for n 12; 24 (mod 36) and n 24. Theorem 3.6 There exists a cyclic 36-cycle systems of K n;n if and only if n 0 (mod 6) and n 18. Lemma 3.7 There is no cyclic 48-cycle system of K n;n for n 12 (mod 48). Lemma 3.8 There is no cyclic 48-cycle system of K n;n for n 36 (mod 48). Theorem 3.9 There exists a cyclic 48-cycle systems of K n;n if and only if n 0(mod 24).

Lemma 3.10 There exists a cyclic 64-cycle systems of K n;n for n 16 (mod 32) and n 48. Lemma 3.11 There is no cyclic 64-cycle system of K n;n for n 8 (mod 32). Lemma 3.12 There is no cyclic 64-cycle system of K n;n for n 24 (mod 32). Theorem 3.13 There exists a cyclic 64-cycle systems of K n;n if and only if n 0(mod 16) and n 32. Lemma 3.14 There exists a cyclic 72-cycle systems of K n;n for n 0 (mod 12) and n 36

Theorem 3.15 There exists a cyclic 72-cycle systems of K n;n if and only if n 0 (mod 12) and n 36. Lemma 3.16 There is no cyclic 80-cycle system of K n;n for n 20 (mod 40). Theorem 3.17 There exists a cyclic 80-cycle systems of K n;n if and only if n 0(mod 40) Lemma 3.18 There exists a cyclic 96-cycle systems of K n;n for n 24 (mod 48) and n 72. Theorem 3.19 There exists a cyclic 96-cycle systems of K n;n if and only if n 0 (mod 24) and n 48. Lemma 3.20 There exists a cyclic 100-cycle systems of K n;n for n 0 (mod 20) and n 60.

Lemma 3.21 There exists a cyclic 100-cycle systems of K n;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 K n;n if and only if n 0 (mod 2m). Lemma 4.2 If there exists a cyclic 50-cycle system of K n;n, then n 0 (mod 20). Lemma 4.3 There exists a cyclic 50-cycle systems of K n;n for n 20; 60 (mod 100) and n 40. Lemma 4.4 There exists a cyclic 50-cycle systems of K n;n for n 40; 80 (mod 100).

Theorem 4.5 There exists a cyclic 50-cycle systems of K n;n if and only if n 0(mod 20) and n 40. Lemma 4.6 If there exists a cyclic 54-cycle system of K n;n, then n 0 (mod 36). Lemma 4.7There exists a cyclic 54-cycle systems of K n;n for n 36; 72 (mod 108). Theorem 4.8 There exists a cyclic 54-cycle systems of K n;n if and only if n 0(mod 36). Lemma 4.9 If there exists a cyclic 90-cycle system of K n;n, then n 0 (mod 60).

Lemma 4.10 There exists a cyclic 90-cycle systems of K n;n for n 0 (mod 60) Theorem 4.11 There exists a cyclic 90-cycle systems of K n;n if and only if n 0(mod 60). Lemma 4.12 If there exists a cyclic 98-cycle system of K n;n, then n 0; 28; 56; 84; 112; 140; (mod 196). Lemma 4.13 There exists a cyclic 98-cycle systems of K n;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.

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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. Acknowledgemen t Acknowledgeme n t

谢 谢 ！