87 research outputs found
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
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
On the minisymposium problem
The generalized Oberwolfach problem asks for a factorization of the complete
graph into prescribed -factors and at most a -factor. When all
-factors are pairwise isomorphic and is odd, we have the classic
Oberwolfach problem, which was originally stated as a seating problem: given
attendees at a conference with circular tables such that the th
table seats people and , find a seating
arrangement over the days of the conference, so that every
person sits next to each other person exactly once.
In this paper we introduce the related {\em minisymposium problem}, which
requires a solution to the generalized Oberwolfach problem on vertices that
contains a subsystem on vertices. That is, the decomposition restricted to
the required vertices is a solution to the generalized Oberwolfach problem
on vertices. In the seating context above, the larger conference contains a
minisymposium of participants, and we also require that pairs of these
participants be seated next to each other for
of the days.
When the cycles are as long as possible, i.e.\ , and , a flexible
method of Hilton and Johnson provides a solution. We use this result to provide
further solutions when and all cycle lengths are
even. In addition, we provide extensive results in the case where all cycle
lengths are equal to , solving all cases when , except possibly
when is odd and is even.Comment: 25 page
On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity
The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2-factor F1 and s copies of a 2-factor F2 such that r+s = v−1 2 . If F1 consists of m-cycles and F2 consists of n cycles, we say that a solution to (m, n)- HWP(v; r, s) exists. The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a solution to (2kx, y)-HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}
On uniformly resolvable -designs
In this paper we consider the uniformly resolvable decompositions of the complete graph minus a 1-factor into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars
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