209 research outputs found
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
Uniformly resolvable decompositions of Kv in 1-factors and 4-stars
If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. A uniformly resolvable {X, Y }-decomposition of the complete graph Kv is an edge decomposition of Kv into exactly r X-factors and s Y -factors. In this article we determine necessary and sufficient conditions for when the complete graph Kv has a uniformly resolvable decompositions into 1-factors and K1,4-factors
Hamilton-Waterloo problem with triangle and C9 factors
The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
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
Substructures in Latin squares
We prove several results about substructures in Latin squares. First, we
explain how to adapt our recent work on high-girth Steiner triple systems to
the setting of Latin squares, resolving a conjecture of Linial that there exist
Latin squares with arbitrarily high girth. As a consequence, we see that the
number of order- Latin squares with no intercalate (i.e., no
Latin subsquare) is at least . Equivalently,
, where is the number
of intercalates in a uniformly random order- Latin square.
In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the
general large-deviation problem for intercalates in random Latin squares, up to
constant factors in the exponent: for any constant we have
and for
any constant we have
.
Finally, we show that in almost all order- Latin squares, the number of
cuboctahedra (i.e., the number of pairs of possibly degenerate
subsquares with the same arrangement of symbols) is of order , which is
the minimum possible. As observed by Gowers and Long, this number can be
interpreted as measuring "how associative" the quasigroup associated with the
Latin square is.Comment: 32 pages, 1 figur
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