Hamiltonian and long paths in bipartite graphs with connectivity

Let G be a graph, $\nu (G)$ the order of G, $\kappa (G)$ the connectivity of G and k a positive integer such that $k\le (\nu (G)?2)/2$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets $V_1$ and $V_2$ is defined to be Hamiltonian-laceable if such that $|V_1|=|V_2|$ and for every pair of vertices $p\in V_1$ and $q\in V_2$, there exists a Hamiltonian path in G with endpoints p and q, or $|V_1|=|V_2|+1$ and for every pair of vertices $p,q\in V_1,p\ne q$, there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition $(X,Y)$. Define $bn(G)$ to be a maximum integer such that $0\le bn(G)<min\{|X|,|Y|\}$ and (1) for each non-empty subset S of X, if $|S|\le |X|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|X|?bn(G)<|S|\le |X|$, then $N(S)=Y$; and (2) for each non-empty subset S of Y, if $|S|\le |Y|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|Y|?bn(G)<|S|\le |Y|$, then $N(S)=X$; and (3) $bn(G)=0$ if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|$ and $bn(G)>0$. In this paper, we show that if $\nu (G)\le 2\kappa (G)+4bn(G)?4$, then G is Hamiltonian-laceable; or if $\nu (G)>6bn(G)?2$, then for every pair of vertices $x\in X$ and $y\in Y$, there is an $(x,y)$-path P in G with $|V(P)|\ge 6bn(G)?2$. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.     

Monochromatic k-edge-connection colorings of graphs

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\ge 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths are colored with a same color. We use $m{c}_k\left(G\right)$ and $um{c}_k\left(G\right)$ to denote the maximum number of colors that ensures $G$ to be monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly monochromatic $k$-edge-connected. In this paper, we first conjecture that for any $k$-edge-connected graph $G$, $m{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+\lfloor \frac{k}{2}\rfloor$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$. We verify the conjecture for $k=2$. We also prove the conjecture for $G=K_{k+1}$ and $G=K_{k,n}$ with $n\ge k\ge 3$. When $G$ is a minimal $k$-edge-connected graph, we give an upper bound of $m{c}_k\left(G\right)$, i.e., $m{c}_k\left(G\right)\le k-1$. For the uniformly monochromatic $k$-edge-connectivity, we prove that for all $k$, $um{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+1$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$.     

颗粒材料三维应力路径下的接触组构特性

颗粒材料的宏观应力变形特征与其微观接触力、组构等紧密相关.一般而言,强接触系统属于颗粒内部体系的传力结构,其对应的组构张量是影响宏观应力性质的重要因素.细观数值方法(如离散单元法)能够反映物理试验的基本规律,并且可以方便地提取宏微观数据来研究颗粒体系的应力变形机制.采用离散单元法(discrete element method,DEM)进行一系列等$p$等$b$应力路径下颗粒材料的真三轴试验,在此基础上研究了三维应力路径下颗粒材料的宏微观力学参数的演化过程、三维组构张量与应力张量多重联系以及强接触体系反映的宏观应力特征.研究表明:颗粒体系偏应力峰值状态和临界状态均存在与加载路径无关的宏微观特征;三维应力路径下组构张量与应力张量存在非共轴性,但其联合不变量演化过程表现出加载路径无关的特征;与弱接触系统的组构张量相比,强接触系统的组构张量更能反映宏观应力张量的特征;强弱接触体系的组构张量对颗粒体系宏观响应的贡献不同,其分界点存在一定取值范围,但采用平均接触力较为简单合理.      

基于FDI收益分成视角的FTA网络演化

本文假定双边自由贸易协定(bilateral free trade agreement,简称FTA)包含着无限制对外直接投资(foreign direct investment,简称FDI),并且通过FDI销售到非FTA伙伴国的收益按照一定比例在母国和东道国之间进行分配。基于Goyal和Joshi^[1],本文构建了FTA网络形成博弈模型。本文发现,FTA网络演化过程分为两个阶段:第一阶段,从空FTA网络到星状FTA网络,存在一条路径使得个体国家福利、世界总福利均改善,在此过程中,国家福利存在不对称性;第二阶段,从星状FTA网络到全连接FTA网络,存在一条路径使得个体国家福利改善,在此过程中,世界总福利不变,国家福利不对称性逐步消除。     

White and some colored marbles as alternative radiation shielding materials for applications

ABSTRACT

In this paper, the radiation shielding parameters such as linear attenuation coefficients (LAC, µ), mass attenuation coefficients (MAC, µ/ρ), effective atomic numbers (Z_eff), effective electron densities (N_eff), half value of layers (HVL), mean free paths (MFP) and buildup factors (exposure (EBF) and energy absorption (EABF)) were investigated for cream (M1), pink (M2), white (M3), maroon (M4) and green (M5) marbles. Attenuation coefficients were measured in the energy region 31.18–661.66 keV photon energies. The values of Z_eff and N_eff were then calculated using these coefficients with logarithmic interpolation method, and HVLs and MFPs were calculated using the values of LAC of marble samples at the same photon energies. The experimental results were compared with the theoretical values obtained from WinXCom program, and good agreements were observed between the experimental and theoretical results. HVLs and MFPs of all marble samples were compared with those of some concretes, glasses and commercial radiation shielding glasses (SCHOTT Co.). The studied marbles were better radiation shielding materials than standard shielding concretes due to lower HVL and MFP values lower than the ordinary concrete. Finally, EBFs and EABFs of the marbles were calculated in the energy region 0.015–1?MeV up to penetration depths of 40 mfps by Geometric Progression method (G-P), and the results were discussed in terms of photon energies and chemical compositions of the marbles.