Decompositions of edge-colored infinite complete graphs into monochromatic paths

An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every $r$-edge colored countably infinite complete $k$-uniform hypergraph can be partitioned into $r$ monochromatic tight paths with distinct colors (a tight path in a $k$-uniform hypergraph is a sequence of distinct vertices such that every set of $k$ consecutive vertices forms an edge), (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ such that the vertex set of every $r$-edge colored countably infinite complete graph can be partitioned into $M$ monochromatic $k^{th}$ powers of paths apart from a finite set (a $k^{th}$ power of a path is a sequence $v_0, v_1, \dots$ of distinct vertices such that $1\le|i-j| \le k$ implies that $v_iv_j$ is an edge), (3.) the vertex set of every $2$-edge colored countably infinite complete graph can be partitioned into $4$ monochromatic squares of paths, but not necessarily into $3$, (4.) the vertex set of every $2$-edge colored complete graph on $\omega_1$ can be partitioned into $2$ monochromatic paths with distinct colors

Towards random uniform sampling of bipartite graphs with given degree sequence

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of {\em semi-regular} degree sequence. The novelty of our approach lays in the construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain explicitly our main contribution and corrected a serious flaw in the cycle decompositio

Comparing weak versions of separability

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8]

Balanced Vertices in Trees and a Simpler Algorithm to Compute the Genomic Distance

This paper provides a short and transparent solution for the covering cost of white-grey trees which play a crucial role in the algorithm of Bergeron {\it et al.}\ to compute the rearrangement distance between two multichromosomal genomes in linear time ({\it Theor. Comput. Sci.}, 410:5300-5316, 2009). In the process it introduces a new {\em center} notion for trees, which seems to be interesting on its own.Comment: 6 pages, submitte

The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $\gamma>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erd\H{o}s-R\'enyi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.Comment: Clarification

A tall space with a small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if kappa(<kappa) = kappa then there is such a space of height kappa(+) with only kappa many isolated points. This implies that there is a locally compact scattered space of height omega(2) with omega(1) isolated points in ZFC, solving an old problem of the first author

Land-Based Exercise and Its Effect on Quality of Life Outcomes in Patients with Knee Osteoarthritis: A Systematic Review

Background: The purpose of this systematic review was to examine effectiveness of land-based exercise interventions for improving quality of life (QOL) of individuals with symptomatic knee osteoarthritis. Methods: A systematic search included PubMed, CINAHL, Scopus, Academic Search Premier, and the Cochrane Database of Systematic Reviews. Inclusion criteria included land-based exercise interventions aimed to improve lower extremity strength and QOL, published since 2000. Exclusion criteria included OA of joints other than the knee, and aquatic-based and surgical interventions. Studies were evaluated using a modified version of the American Academy of Cerebral Palsy and Developmental Medicine (AACPDM) methodology score. Results: Level of evidence for 11 studies ranged from I strong (I-S) to IV. There were 1200 total participants 57.5 to 69.8 years of age with a mean of 64.5. Three common treatment groups were used; exercise, yoga, and education. Twenty outcome measures were used with the Western Ontario and McMaster Universities Arthritis Index (WOMAC), Short Form 36 (SF-36) and strength being most common. Discussion: Inconsistencies in reporting outcome measures and their subgroups, data, and statistical analyses prevented further data analysis to compare individual intervention effectiveness. However, many articles reported significant improvements in varying QOL subgroups and strength measurements. Conclusions: This review suggests treatment approaches involving physical activity will be beneficial across all levels of the International Classification of Functioning Disability and Health (ICF), including QOL