Spanning trees in sparse expanders

Given integers $n\ge \Delta\ge 2$, let $\mathcal{T}(n, \Delta)$ be the collection of all $n$-vertex trees with maximum degree at most $\Delta$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible spectral gap condition forcing an $(n, d,\lambda)$-graph to be $\mathcal{T}(n, \Delta)$-universal, namely, it contains all members of $\mathcal{T}(n, \Delta)$ as a subgraph simultaneously. In this paper we show that for all $\Delta\in \mathbb{N}$ and sufficiently large $n$, every $(n, d,\lambda)$-graph with $$\lambda\le\frac{d}{2\Delta^{5\sqrt{\log n}}}$$ is $\mathcal{T}(n, \Delta)$-universal. As an immediate corollary, this implies that Alon's ingenious construction of triangle-free sparse expander is $\mathcal{T}(n, \Delta)$-universal, which provides an explicit construction of such graphs and thus solves a question of Johannsen, Krivelevich and Samotij. Our main result is formulated under a much more general context, namely, the $(n,d)$-expanders. More precisely, we show that there exist absolute constants $C,c>0$ such that the following statement holds for sufficiently large integer $n$. $(1)$ For all $\Delta\in \mathbb{N}$, every $(n, \Delta^{5\sqrt{\log n}})$-expander is $\mathcal{T}(n, \Delta)$-universal. $(2)$ For all $\Delta\in \mathbb{N}$ with $\Delta \le c\sqrt{n}$, every $(n, C\Delta n^{1/2})$-expander is $\mathcal{T}(n, \Delta)$-universal. Both results significantly improve a result of Johannsen, Krivelevich and Samotij, and have further implications in locally sparse expanders and Maker-Breaker games that also improve previously known results drastically.Comment: 27 pages, 4 figures, comments are welcom

On powers of Hamilton cycles in Ramsey-Tur\'{a}n Theory

We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $\mu>0$, there exist $\alpha>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}+\mu\right)n$ and $\alpha(G)\leq \alpha n$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.Comment: 19 pages, 4 figure

Rainbow Hamilton cycle in hypergraph systems

R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di proved that every $n$-vertex $k$-graph $H$, $k\geq3, \gamma>0$ and $n$ is sufficiently large, with $\delta_{k-1}(H)\geq(1/2+\gamma)n$ contains a tight Hamilton cycle, which can be seen as a generalization of Dirac's theorem in hypergraphs. In this paper, we extend this result to the rainbow setting as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, a $k$-graph $G$ on $V$ is rainbow if $E(G)\subseteq\bigcup_{i\in[m]}E(H_i)$ and $|E(G)\cap E(H_i)|\leq 1$ for $i\in[m]$. Then we show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for $i\in[n]$, then there exists a rainbow tight Hamilton cycle.Comment: 20 pages,5 figure

On the Population Monotonicity of Independent Set Games

An independent set game is a cooperative game defined on graphs and dealing with profit sharing in maximum independent set problems. A population monotonic allocation scheme is a rule specifying how to share the profit of each coalition among its participants such that every participant is better off when the coalition expands. In this paper, we provide a necessary and sufficient characterization for population monotonic allocation schemes in independent set games. Moreover, our characterization can be verified efficiently

Production of human blood group B antigen epitope conjugated protein in Escherichia coli and utilization of the adsorption blood group B antibody

Additional file 1: Table S1. List of constructed plasmids, strains and primers used in the study. Figure S1. MALDI-TOF detection of MBPmut (a) and MBPmut-OPS (b)

A genome-wide association study of severe asthma exacerbations in Latino children and adolescents

Severe asthma exacerbations are a major cause of school absences and healthcare costs in children, particularly those in high-risk racial/ethnic groups. To identify susceptibility genes for severe asthma exacerbations in Latino children and adolescents, we conducted a meta-analysis of genome-wide association studies (GWAS) in 4010 Latino youth with asthma in four independent cohorts, including 1693 Puerto Ricans, 1019 Costa Ricans, 640 Mexicans, 256 Brazilians, and 402 members of other Latino subgroups. We then conducted methylation quantitative trait locus (mQTL), expression quantitative trait locus (eQTL), and expression quantitative trait methylation (eQTM) analyses to assess whether the top SNP in the meta-analysis is linked to DNA methylation and gene expression in nasal (airway) epithelium in separate cohorts of Puerto Rican and Dutch children and adolescents. In the meta-analysis of GWAS, a SNP in FLJ22447 (rs2253681) was significantly associated with 1.55 increased odds of severe asthma exacerbations (95% confidence interval=1.34 to 1.79, p=6.3×10-9). This SNP was significantly associated with DNA methylation of a CpG site (cg25024579) at the FLJ22447 locus, which was in turn associated with increased expression of KCNJ2-AS1 in nasal airway epithelium from Puerto Rican children and adolescents (β=0.10, p=2.18×10-7). Thus, SNP rs2253681 was significantly associated with both DNA methylation of a cis-CpG in FLJ22447 and severe asthma exacerbations in Latino youth. This may be partly explained by changes in airway epithelial expression of a gene recently implicated in atopic asthma in Puerto Rican children and adolescents (KCNJ2-AS1)

On-line scheduling of small open shops

We investigate theprobl4 ofon-l.E schedulw4 open shops of two and three machines with an objective of minimizing theschedul makespan. We #rst propose a 1.848-competitive permutationalmutati for the non-preemptiveschedulmp problu of two machines and show that no permutationalrmutat can be better than 1.754-competitive.Secondlo wedevel. a (27=19)-competitiveal)-comp for the preemptive schedulve problu of three machines, which is most competitive. ? 2001El1C8.7 Science B.V.Al rights reserved

HH-factors in graphs with small independence number

Let $H$ be an $h$-vertex graph. The vertex arboricity $ar(H)$ of $H$ is the least integer $r$ such that $V(H)$ can be partitioned into $r$ parts and each part induces a forest in $H$. We show that for sufficiently large $n\in h\mathbb{N}$, every $n$-vertex graph $G$ with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+o(1)\right)n, \left(\frac{1}{2}+o(1)\right)n\right\}$ and $\alpha(G)=o(n)$ contains an $H$-factor, where $f(H)=2ar(H)$ or $2ar(H)-1$. The result can be viewed an analogue of the Alon--Yuster theorem \cite{MR1376050} in Ramsey--Tur\'{a}n theory, which generalises the results of Balogh--Molla--Sharifzadeh~\cite{MR3570984} and Knierm--Su~\cite{MR4193066} on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs $H$ which are not cliques.Comment: 25 pages, 1 figur

Experimental Research on Variable Parameter Forming Process for Forming Specimen of TC4 Titanium Alloy by Selective Laser Melting

To optimize the microstructure and properties of TC4 specimens formed by selective laser melting (SLM), the test program of formed specimens by the variable parameter forming process (VPFP) was designed based on the quantitative parameter forming process (QPFP). The purpose of this study is to explore the influence of the VPFP on the surface morphology, tensile properties, and microstructure of the specimens. The test results show that the surface morphology and tensile properties of the specimens were better formed by the VPFP. The internal holes of the specimens formed by the VPFP were small in volume and occupied a relatively small proportion, and the density could reach 99.7%. When the laser power was 300 W–260 W and equally divided into six hierarchies, the tensile strength could reach 1185.214 MPa by VPFP, but the elongation had no obvious change. The number of secondary acicular martensite α’ phases was decreased in the microstructure of the specimens formed with VPFP. With the superposition of the hierarchy, the length of the primary acicular martensite α’ phase became shorter, the width became larger, and the width of the columnar crystal β phase became smaller. The VPFP is used to change the inherent method of forming specimens with the same parameters, which provides a new idea for SLM-forming structures; the test provides data and yields a theoretical research basis for forming the specimens process method