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].     

A New Approach for the Synthesis of Miktoarm Star Polymers Through a Combination of Thiol–Epoxy “Click” Chemistry and ATRP/Ring‐Opening Polymerization Techniques

A new approach was developed for synthesis of certain A₃B₃‐type of double hydrophilic or amphiphilic miktoarm star polymers using a combination of “grafting onto” and “grafting from” methods. To achieve the synthesis of desired miktoarm star polymers, acetyl protected poly(ethylene glycol) (PEG) thiols (M_n = 550 and 2000 g mol^?1) were utilized to generate A₃‐type of homoarm star polymers through an in situ protective group removal and a subsequent thiol–epoxy “click” reaction with a tris‐epoxide core viz. 1,1,1‐tris(4‐hydroxyphenyl)ethane triglycidyl ether. The secondary hydroxyl groups generated adjacent to the core upon the thiol–epoxy reaction were esterified with α‐bromoisobutyryl bromide to install atom transfer radical polymerization (ATRP) initiating sites. ATRP of N‐isopropylacrylamide (NIPAM) using the three‐arm star PEG polymer fitted with ATRP initiating sites adjacent to the core afforded A₃B₃‐type of double hydrophilic (PEG)₃[poly(N‐isopropylacrylamide)] (PNIPAM)₃ miktoarm star polymers. Furthermore, the generated hydroxyl groups were directly used as initiator for ring‐opening polymerization of ε‐caprolactone to prepare A₃B₃‐type of amphiphilic (PEG)₃[poly(ε‐caprolactone)]₃ miktoarm star polymers. The double hydrophilic (PEG)₃(PNIPAM)₃ miktoarm star polymers showed lower critical solution temperature around 34 °C. The preliminary transmission electron microscopy analysis indicated formation of self‐assembly of (PEG)₃(PNIPAM)₃ miktoarm star polymer in aqueous solution. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 146–156     

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.     

Miktoarm star‐shaped poly(lactic acid) copolymer: Synthesis and stereocomplex crystallization behavior

The miktoarm star‐shaped poly(lactic acid) (PLA) copolymer, (PLLA)₂‐core‐(PDLA)₂, was synthesized via stepwise ring‐opening polymerization of lactide with dibromoneopentyl glycol as the starting material. ¹H NMR and FTIR spectroscopy proved the feasibility of synthetic route and the successful preparation of star‐shaped PLA copolymers. The results of FTIR spectroscopy and XRD showed that the stereocomplex structure of the copolymer could be more perfect after solvent dissolution treatment. Effect of chain architectures on crystallization was investigated by studying the nonisothermal and isothermal crystallization of the miktoarm star‐shaped PLA copolymer and other stereocomplexes. Nonisothermal differential scanning calorimetry and polarizing optical microscopy tests indicated that (PLLA)₂‐core‐(PDLA)₂ exhibited the fastest formation of a stereocomplex in a dynamic test due to its special structure. In isothermal crystallization tests, the copolymer exhibited the fast crystal growth rate and the most perfect crystal morphology. The results reveal that the unique molecular structure has an important influence on the crystallization of the miktoarm star‐shaped PLA copolymer. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 814–826     

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$\chi_{\rm sat}（G）\leqslant 9$。