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?
•
The Hamilton-Waterloo problem asks for a 2factorization of the complete graph K n (n is
odd) or K n  I (n is even and I 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.
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).
HW(4;1,0;4,4)
1
0
1
0
1
0
j
3
3
2
3
HW(5;2,0;5,5)
0
4
1
3
2
2
2
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.
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 HamiltonWaterloo 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 HamiltonWaterloo problem: the case of triangle-factors and
one Hamilton cycle, J. Combin. Designs. 17 (2009)
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).
Method
• Let Z4×Zk be the vertex set of Kn. We write
{0}  Z k  A  {ai : i  0,1,, k  1},
{1}  Z k  B  {bi : i  0,1,, k  1},
{2}  Z k  C  {ci : i  0,1, , k  1},
{3}  Z k  D  {di : i  0,1, , k  1}
for simplicity of description. All the subscripts are
taken modulo k.
Method
•
For 0  d  k 1, define sets of edges
( AB)d  {(ai , bi  d ) : i  0,1,
, k 1}
( BC )d , (CD)d , ( DA)d , ( AC ) d , ( BD)d 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]∪ ( AB
∪ )d ( BC
∪ )d
( DA)∪
( AC. )d ( BD)d
∪ (CD)∪
d
d
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 ( AB) p  ( BC )q  (CD)r  ( DA) s
induces an HC of K n , as well as edge sets
( AB) p  ( BD) q  ( DC ) r  (CA) s and ( AC ) p  (CB)q  ( BD) r
( DA) s .
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 ( AB) p  ( BC )q  (CD)r  ( DA) s induces an
4-cycle factor of K n , as well as edge sets
( AC ) p  (CB)q  ( BD) r ( DA) s , and ( AB) p  ( BD) q  ( DC ) r
(CA)s ,
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.
谢谢