1. The Hamilton-Waterloo problem for Hamilton cycles and 4-cycle factors Hongchuan Lei, Hao Shen, Ming Luo Shanghai Jiao Tong University

2. What is Hamilton-Waterloo Problem? The Hamilton-Waterloo problem asks for a 2- factorization of the complete graph (n is odd) or (n is even and is a 1-factor of K n ) in which r of its 2-factors are isomorphic to a given 2-factor R and s of the its 2-factors are isomorphic to another given 2-factor S.

3. If the components of R are cycles of length m and the components of S are cycles of length k, then the corresponding Hamilton-Waterloo problem is denoted by HW(n;r,s;m,k).

4. HW(4;1,0;4,4) HW(5;2,0;5,5)

5. The first paper on this topic deals with the HW(n;r,s;m,k) when (m,k) ∈ {(3,5),(3,15),(5,15)}. P. Adams, E.J. Billinton, D.E. Bryant, S.I. El-Zanati, On the Hamilton-Waterloo problem, Graph Combin. A recent article completely solves the HW(n;r,s;3,4) only with a few possible exceptions when n=24 and 48. P. Danziger, G. Quattrocchi, B. Stevens, The Hamilton-Waterloo problem for cycle sizes 3 and 4, J. Combin. Designs. Papers on this topic

6. The following 3 papers completely solved the HW(n;r,s;n,3) with only 14 possible exceptions. P. Horak, R. Nedela, and A. Rosa, The Hamilton- Waterloo problem: the case of Hamilton cycles and triangle- factors, Discrete Math 284 (2004), J. H. Dinitz, A. C. H. Ling, The Hamilton-Waterloo problem with triangle-factors and Hamilton cycles: The case n ≡ 3 (mod 18), J. Combin. Math. Combin. Comput. In press. J.H. Dinitz, A. C. H. Ling, The Hamilton-Waterloo problem: the case of triangle-factors and one Hamilton cycle, J. Combin. Designs. 17 (2009) Papers on this topic

7. The special case of Hamilton-Waterloo problem that we will deal with is the case R is a Hamilton cycle and S is a 4-cycle factor (consisted of cycles of length 4). Our Research

8. Let Z 4 ×Z k be the vertex set of K n. We write for simplicity of description. All the subscripts are taken modulo k. Method

9. For, define sets of edges are the similar. [A] is the edge set of the complete graph on A. Then the edge set of the complete graph K n is [A] ∪ [B] ∪ [C] ∪ [D] ∪ ∪ ∪ ∪ ∪ ∪. Method

10. Lemma 1. Let -k+1 ≤ p,q,r,s ≤ k-1 be integers such that p+q+r+s and k are relatively prime then the set of edges induces an HC of, as well as edge sets and Method

11. Lemma 2. Let -k+1 ≤ p,q,r,s ≤ k-1 be integers such that p+q+r+s ≡ 0 (mod k) then the set of edges induces an 4-cycle factor of, as well as edge sets and Method

12. Theorem There is a solution to the Hamilton- Waterloo problem on n points with Hamilton cycles and 4-cycle factors for positive integer n ≡ 0 (mod 4) and all possible numbers of Hamilton cycles. Our Result

13. 谢谢