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The Hamilton-Waterloo problem for Hamilton cycles and 4-cycle factors

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

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The Hamilton-Waterloo problem for Hamilton cycles and 4-cycle factors

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  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 Kn) in which r of its 2-factors are isomorphic to a given 2-factorR 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. Papers on this topic 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.

  6. Papers on this topic 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)

  7. Our Research 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).

  8. Method • Let Z4×Zk be the vertex set of Kn. We write for simplicity of description. All the subscripts are taken modulo k.

  9. Method • 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 Knis [A]∪[B]∪[C]∪[D]∪ ∪ ∪ ∪ ∪ ∪ .

  10. Method 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

  11. Method 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

  12. Our Result • 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.

  13. 谢谢

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