Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Linearly χ-bounding (P6, C4)-free graphs*

Given two graphs and , a graph is -free if it contains no induced subgraph isomorphic to or . Let and be the path on vertices and the cycle on vertices, respectively. In this paper we show that for any -free graph it holds that , where and are the chromatic number and clique number of , respectively. Our bound is attained by several graphs, for instance, the 5-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all -critical -free graphs other than (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear -binding functions for several graph classes. Our proof is based on a novel structure theorem on -free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time -approximation algorithm for coloring -free graphs. Our algorithm computes a coloring with colors for any -free graph in time.     

硬X射线微束掠入射实验的控制和数据采集系统及其应用

在国内发展了硬X射线微束掠入射实验方法，并将此具有微米级高空间分辨率的方法应用于纳米厚度薄膜的微区分析。该实验方法对分析样品表面或薄膜在微小区域的不均匀组分、结构、厚度、粗糙度和表面元素化学价态等信息具有重要意义。基于X射线全反射原理，以微聚焦实验站的高通量、能量可调的单色微束X射线为基础，通过集成运动控制、光强探测、衍射和荧光探测，设计了掠入射实验方法的控制和数据采集系统。此系统采用分布式控制结构，并基于Experimental Physics and Industrial Control System （EPICS）环境设计SPEC控制软件。通过建立SPEC和EPICS的访问通道，实现SPEC软件对EPICS平台上设备的控制和数据获取。在所设计的控制和数据采集系统中，运动控制系统控制多维样品台电机的运动，实现定位样品位置和调节掠入射角；光强探测系统则监测样品出射光强度，通过样品台运动控制和光强探测的联控，实现样品台的扫描定位控制；通过衍射和荧光探测系统获取不同入射深度下样品的衍射峰强度和荧光计数。此外，为准确控制掠入射角角度，必须确定样品平面与X射线平行的零角度位置，对此给出一种自动定位零角度的方法，编写了该方法的控制算法，设计了相应的控制软件。零角度自动化定位的扫描结果表明，实验系统微区分析的空间分辨率达到2.8 μm，零角度定位精度小于±0.01°。利用该系统在上海光源微聚焦实验站首次实现了具有自动化准确控制零角度的微束掠入射X射线衍射和荧光同步表征的实验方法，实验中被测样品为10 nm Au/Cr/Si薄膜材料，Si基底最上层为10 nm厚的Au薄膜，其间为一层很薄的Cr粘附层。在不同掠入射角下测量样品的衍射信号，获取不同入射深度下样品的衍射峰强度，并实现在同一掠入射角下，同步采集样品的荧光计数信号，从而确定了样品表层的相结构信息以及荧光信号强度与入射角关系，实现了对纳米厚度薄膜在微小区域的相结构和组分分析。此外，通过该技术能够选取荧光计数最大值对应的入射角度，有助于提高后续发展的低浓度样品掠入射X射线吸收近边结构实验方法的信噪比。     

An infinite family of subcubic graphs with unbounded packing chromatic number

Recently, Balogh et al. (2018) answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree 3. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.     

Δ(G)=5的2-连通外平面图的邻点可区别全染色

Smarandachely邻点可区别全染色是指相邻点的色集合互不包含的邻点可区别全染色，是对邻点可区别全染色条件的进一步加强。本文研究了平面图的Smarandachely邻点可区别全染色，即根据2-连通外平面图的结构特点，利用分析法、数学归纳法，刻画了最大度为5的2-连通外平面图的Smarandachely邻点可区别全色数。证明了：如果$G$是一个$\Delta （G）=5$的2-连通外平面图，则$\chi_{\rm sat}（G）\leqslant 9$。