49 research outputs found
Merging Combinatorial Design and Optimization: the Oberwolfach Problem
The Oberwolfach Problem , posed by Gerhard Ringel in 1967, is a paradigmatic Combinatorial Design problem asking whether the complete graph decomposes into edge-disjoint copies of a -regular graph of order . In Combinatorial Design Theory, so-called difference methods represent a well-known solution technique and construct solutions in infinitely many cases exploiting symmetric and balanced structures. This approach reduces the problem to finding a well-structured -factor which allows us to build solutions that we call - or -rotational according to their symmetries. We tackle by modeling difference methods with Optimization tools, specifically Constraint Programming () and Integer Programming (), and correspondingly solve instances with up to within . In particular, we model the -rotational method by solving in cascade two subproblems, namely the binary and group labeling, respectively. A polynomial-time algorithm solves the binary labeling, while tackles the group labeling. Furthermore, we prov ide necessary conditions for the existence of some -rotational solutions which stem from computational results. This paper shows thereby that both theoretical and empirical results may arise from the interaction between Combinatorial Design Theory and Operation Research
A constructive solution to the Oberwolfach Problem with a large cycle
For every -regular graph of order , the Oberwolfach problem
asks whether there is a -factorization of ( odd) or minus a
-factor ( even) into copies of . Posed by Ringel in 1967 and
extensively studied ever since, this problem is still open. In this paper we
construct solutions to whenever contains a cycle of length greater
than an explicit lower bound. Our constructions combine the
amalgamation-detachment technique with methods aimed at building
-factorizations with an automorphism group having a nearly-regular action on
the vertex-set.Comment: 11 page
Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor
We prove that for and an absolute constant , if and is a random subset of where
each is included in independently with probability for
each , then asymptotically almost surely there is an order-
Latin square in which the entry in the th row and th column lies in
. The problem of determining the threshold probability for the
existence of an order- Latin square was raised independently by Johansson,
by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides
an upper bound which is tight up to a factor of and strengthens the
bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous
results for Steiner triple systems and -factorizations of complete graphs,
and moreover, we show that each of these thresholds is at most the threshold
for the existence of a -factorization of a nearly complete regular bipartite
graph.Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the
AM
Post-critical set and non existence of preserved meromorphic two-forms
We present a family of birational transformations in depending on
two, or three, parameters which does not, generically, preserve meromorphic
two-forms. With the introduction of the orbit of the critical set (vanishing
condition of the Jacobian), also called ``post-critical set'', we get some new
structures, some "non-analytic" two-form which reduce to meromorphic two-forms
for particular subvarieties in the parameter space. On these subvarieties, the
iterates of the critical set have a polynomial growth in the \emph{degrees of
the parameters}, while one has an exponential growth out of these subspaces.
The analysis of our birational transformation in is first carried out
using Diller-Favre criterion in order to find the complexity reduction of the
mapping. The integrable cases are found. The identification between the
complexity growth and the topological entropy is, one more time, verified. We
perform plots of the post-critical set, as well as calculations of Lyapunov
exponents for many orbits, confirming that generically no meromorphic two-form
can be preserved for this mapping. These birational transformations in ,
which, generically, do not preserve any meromorphic two-form, are extremely
similar to other birational transformations we previously studied, which do
preserve meromorphic two-forms. We note that these two sets of birational
transformations exhibit totally similar results as far as topological
complexity is concerned, but drastically different results as far as a more
``probabilistic'' approach of dynamical systems is concerned (Lyapunov
exponents). With these examples we see that the existence of a preserved
meromorphic two-form explains most of the (numerical) discrepancy between the
topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure
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
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