The Hamilton-Waterloo problem for Hamilton

cycles and 4-cycle factors

Hongchuan Lei, Hao Shen, Ming LuoShanghai Jiao Tong University

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-factor R and s of the its 2-factors are isomorphic to another given 2-factor S.

nKnK I I

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

3 2

1

0

3 2

10

3 2

10

j

3 2

10

HW(4;1,0;4,4)

HW(5;2,0;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

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

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

• Let Z4×Zk be the vertex set of Kn. We write

for simplicity of description. All the subscripts are taken modulo k.

Method

{0} { : 0,1, , 1},

{1} { : 0,1, , 1},

{2} { : 0,1, , 1},

{3} { : 0,1, , 1}

k i

k i

k i

k i

Z A a i k

Z B b i k

Z C c i k

Z D d i k

• 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 Kn is [A] [B] [C] [D] ∪ ∪ ∪ ∪ ∪ ∪ .∪ ∪ ∪

Method

( ) {( , ) : 0,1, , 1}d i i dAB a b i k

0 1d k

( ) ,dBC ( ) ,dCD ( ) ,dDA ( ) ,dAC ( )dBD

( )dAB ( )dBC

( )dCD ( )dBD( )dAC( )dDA

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

( ) ( ) ( ) ( )p q r sAB BC CD DA nK

( ) ( ) ( ) ( )p q r sAB BD DC CA ( ) ( ) ( )p q rAC CB BD ( ) .sDA

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

( ) ( ) ( ) ( )p q r sAB BC CD DA

nK( ) ( ) ( )p q rAB BD DC ( ) ( ) ( )p q rAC CB BD ( ) ,sDA

( ) ,sCA

• 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

谢谢