203 research outputs found
Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles
It is conjectured that for every pair of odd integers greater than
2 with , there exists a cyclic two-factorization of
having exactly factors of type and all the
others of type . The authors prove the conjecture in the affirmative
when and .Comment: 31 page
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
A survey on constructive methods for the Oberwolfach problem and its variants
The generalized Oberwolfach problem asks for a decomposition of a graph
into specified 2-regular spanning subgraphs , called factors.
The classic Oberwolfach problem corresponds to the case when all of the factors
are pairwise isomorphic, and is the complete graph of odd order or the
complete graph of even order with the edges of a -factor removed. When there
are two possible factor types, it is called the Hamilton-Waterloo problem.
In this paper we present a survey of constructive methods which have allowed
recent progress in this area. Specifically, we consider blow-up type
constructions, particularly as applied to the case when each factor consists of
cycles of the same length. We consider the case when the factors are all
bipartite (and hence consist of even cycles) and a method for using circulant
graphs to find solutions. We also consider constructions which yield solutions
with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series.
23 pages, 2 figure
On the Hamilton-Waterloo problem for bipartite 2-factors
Given two 2-regular graphs F1 and F2, both of order n, the Hamilton-Waterloo Problem for F1 and F2 asks for a factorization of the complete graph Kn into a1 copies of F1, a2 copies of F2, and a 1-factor if n is even, for all nonnegative integers a1 and a2 satisfying a1+a2=?n-12?. We settle the Hamilton-Waterloo Problem for all bipartite 2-regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2-regular graph of the same order
Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem
In this paper, we formally introduce the concept of a row-sum matrix over an
arbitrary group . When is cyclic, these types of matrices have been
widely used to build uniform 2-factorizations of small Cayley graphs (or,
Cayley subgraphs of blown-up cycles), which themselves factorize complete
(equipartite) graphs.
Here, we construct row-sum matrices over a class of non-abelian groups, the
generalized dihedral groups, and we use them to construct uniform
-factorizations that solve infinitely many open cases of the
Hamilton-Waterloo problem, thus filling up large parts of the gaps in the
spectrum of orders for which such factorizations are known to exist
Bipartite 2-factorizations of complete multipartite graphs
It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorization of K into F; except that there is no factorization of K into F when F is the union of two disjoint 6-cycles
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
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