251 research outputs found
The directed Oberwolfach problem with variable cycle lengths: a recursive construction
The directed Oberwolfach problem OP asks whether the
complete symmetric digraph , assuming , admits a
decomposition into spanning subdigraphs, each a disjoint union of directed
cycles of lengths . We hereby describe a method for
constructing a solution to OP given a solution to
OP, for some , if certain conditions on
are satisfied. This approach enables us to extend a solution
for OP into a solution for
OP, as well as into a solution for
OP, where denotes copies of 2, provided is sufficiently large.
In particular, our recursive construction allows us to effectively address
the two-table directed Oberwolfach problem. We show that OP has
a solution for all , with a definite exception of
and a possible exception in the case that , is even,
and . It has been shown previously that OP has
a solution if is odd, and that OP has a solution if and
only if .
In addition to solving many other cases of OP, we show that when , OP has a solution if and
only if
A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs
The Hamilton-Waterloo problem asks for which and the complete graph
can be decomposed into copies of a given 2-factor and
copies of a given 2-factor (and one copy of a 1-factor if is even).
In this paper we generalize the problem to complete equipartite graphs
and show that can be decomposed into copies of a
2-factor consisting of cycles of length ; and copies of a 2-factor
consisting of cycles of length , whenever is odd, ,
and . We also give some more general
constructions where the cycles in a given two factor may have different
lengths. We use these constructions to find solutions to the Hamilton-Waterloo
problem for complete graphs
Two Problems of Gerhard Ringel
Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel\u27s research: the Hamilton-Waterloo Problem, and the problem of R-Sequences.
The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where m=3 and n=3x. This problem is settle for odd v, except for a finite number of x values. When v is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations.
Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K(n:m) and shows that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r≠1, gcd(x,z)=gcd(y,z)=1 and xyz≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs.
Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups
Sectionable terraces and the (generalised) Oberwolfach problem
AbstractThe generalised Oberwolfach problem requires v people to sit at s round tables of sizes l1,l2,…,ls (where l1+l2+⋯+ls=v) for successive meals in such a way that each pair of people are neighbours exactly λ times. The problem is denoted OP(λ;l1,l2,…,ls) and if λ=1, which is the original problem, this is abbreviated to OP(l1,l2,…,ls). It was known in 1892, though different terminology was then used, that a directed terrace with a symmetric sequencing for the cyclic group of order 2n can be used to solve OP(2n+1). We show how terraces with special properties can be used to solve OP(2;l1,l2) and OP(l1,l1,l2) for a wide selection of values of l1, l2 and v. We also give a new solution to OP(2;l,l) that is based on Z2l−1. Solutions to the problem are also of use in the design of experiments, where solutions for tables of equal size are called resolvable balanced circuit Rees neighbour designs
The generalised Oberwolfach problem
We prove that any quasirandom dense large graph in which all degrees are
equal and even can be decomposed into any given collection of two-factors
(2-regular spanning subgraphs). A special case of this result gives a new
solution to the Oberwolfach problem.Comment: 32 pages, 4 figure
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Combinatorial Optimization
This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th
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
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Algebraic Statistics
Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research
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