123 research outputs found
The existence of Ck-factorizations of K2n − F
AbstractA necessary condition for the existence of a Ck-factorization of K2n − F is that k divides 2n. It is known that neither K6 − F nor K12 − F admit a C3-factorization. In this paper we show that except for these two cases, the necessary condition is also sufficient
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
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
Partial triple systems and edge colourings
AbstractA partial triple system of order v, PT(v), is pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V (called triples) such that each 2-subset of V is contained in at most one triple. A maximum partial triple system of order v, MPT(v), is a PT(v), (V, B), such that for any other PT(v), (V, C), we have |C| ⪕|B|. Several authors have considered the problem of embedding PT(v) and MPT(v) in systems of higher order. We complete the proof, begun by Mendelsohn and Rosa [6], that an MPT(u) can be embedded in an MPT(v) where v is the smallest value in each congruence class mod 6 with v ⩾ 2u. We also consider a general problem concerning transversals of minimum edge-colourings of the complete graph
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
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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