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Cyclic m -Cycle Systems of K n, n for m 100

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.

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Cyclic m -Cycle Systems of K n, n for m 100

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  1. Cyclic m-Cycle Systems of K n, nfor 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

  2. 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;kif 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 Zvis 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.

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

  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;nfor 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;nfor 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;nfor all integers m 100. ，

  5. 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∈Zv 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)-

  6. -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;nbe 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 (Kn;n;Cm)-DS, then the cycles form a cyclic m-cycle system of Kn;nwhere di is the type of Bi.

  7. Theorem 2.5([9]) There exists a cyclic m-cycle system of Kn;nif 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;nfor 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;nfor 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;nif and only if n 0 (mod m/2 ).

  8. 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;nif 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;nfor n 2 (mod 4). Theorem 2.11([8]) There exists a cyclic 16-cycle systems of Kn;nif and only if n 0 (mod 8). Theorem 2.12([9]) There exists a cyclic 18-cycle systems of Kn;nif and only if n 0 (mod 12) ．

  9. 3 Cyclic m-Cycle System of K n;nfor 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;nif and only if n 0 (mod m/2 ). Lemma 3.2 There exists a cyclic 32-cycle systems of Kn;nfor n 8 (mod 16), and n > 8. Theorem 3.3 There exists a cyclic 32-cycle systems of Kn;nif and only if n 0 (mod 8) and n 16 Lemma 3.4 There exists a cyclic 36-cycle systems of Kn;nfor n 6; 30 (mod 36) and n 30 .

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

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

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

  13. Lemma 3.21 There exists a cyclic 100-cycle systems of Kn;nfor n 10 (mod 20) and n 70. Theorem 3.22 There exists a cyclic 100-cycle systems of K n;nif and only if n 0 (mod 10) and n 50.

  14. 4 Cyclic m-Cycle System of K n;nfor 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;nif 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;nfor n 20; 60 (mod 100) and n 40. Lemma 4.4 There exists a cyclic 50-cycle systems of Kn;nfor n 40; 80 (mod 100).

  15. Theorem 4.5 There exists a cyclic 50-cycle systems of Kn;nif 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;nfor n 36; 72 (mod 108). Theorem 4.8 There exists a cyclic 54-cycle systems of Kn;nif 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).

  16. Lemma 4.10 There exists a cyclic 90-cycle systems of Kn;nfor n 0 (mod 60) Theorem 4.11 There exists a cyclic 90-cycle systems of Kn;nif 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;nfor 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.

  17. 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

  18. [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.

  19. Acknowledgement 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

  20. 谢 谢 ！

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