The number of Hamiltonian decompositions of regular graphs

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K _n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kühn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r ^(1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K _n is \({n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}\).     

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rödl, Erd?s-Hajnal, Prömel-Rödl, Nikiforov, Chung-Graham, and ?uczak-Rödl. The proofs are based on a simple lemma (generalizing one by Graham, Rödl, and Ruciński) that can be used as a replacement for Szemerédi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.     

Low-temperature metallic alloying of copper and silver nanoparticles with gold nanoparticles through digestive ripening

We describe a remarkable and simple alloying procedure in which noble metal intermetallic nanoparticles are produced in gram quantities via digestive ripening. This process involves mixing of separately prepared colloids of pure Au and pure Ag or Cu particles and then heating in the presence of an alkanethiol under reflux. The result after 1 h is alloy nanoparticles. Particles synthesized according to this procedure were characterized by UV-vis spectroscopy, EDX analysis, and high-resolution electron microscopy, the results of which confirm the formation of alloy particles. The particles of 5.6+/-0.5 nm diameter for Au/Ag and 4.8+/-1.0 nm diameter for Cu/Au undergo facile self-assembly to form 3-D superlattice ordering. It appears that during this digestive ripening process, the organic ligands display an extraordinary chemistry in which atom transfer between atomically pure copper, silver, and gold metal nanoparticles yields monodisperse alloy nanoparticles.     

Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations

We study the spreading of viscous drops on a solid substrate, taking into account the effects of thermal fluctuations in the fluid momentum. A nonlinear stochastic lubrication equation is derived and studied using numerical simulations and scaling analysis. We show that asymptotically spreading drops admit self-similar shapes, whose average radii can increase at rates much faster than these predicted by Tanner's law. We discuss the physical realizability of our results for thin molecular and complex fluid films, and predict that such phenomenon can in principal be observed in various flow geometries.     

Phosphorus-based nanothermites: A new generation of energetic materials

Thermites are energetic materials that are classically made of a transition metal oxide mixed with a reducing metal. Contrary to explosives, thermites do not detonate because their combustion is relatively slow and their reaction by-products are often solid or liquid. The use of nanoparticles to prepare superthermites is very promising. The dramatic changes observed in their reactivity were reported by numerous recent papers dealing with the use of aluminium as fuel.Red phosphorus is widely used in pyrotechnic devices. Highly explosive compositions are classically obtained by mixing this substance with strong oxidizers such as oxygenated potassium salts (chlorate, nitrate). But to our knowledge, the idea to prepare P-nanothermites with metallic oxide nanoparticles was never reported before. In order to illustrate this new concept of energetic formulations, P-nanothermites were prepared from nickel oxide (NiO), iron oxide (Fe₂O₃), and copper oxide (CuO) nanopowders. The reactivity of these compositions was studied by thermogravimetric analysis, impact and friction tests, electrostatic discharge and high-speed cinematography. P-nanothermites are very insensitive to thermal and impact stress. Their combustion rates strongly depend on the nature of the oxide (NiO <Fe₂O₃?CuO). The SEM observation of the microstructure of the residues produced by the combustion clearly indicates that they were formed by the solidification of molten phases. In other words, the energy released by the combustion of P-nanothermites provokes the melting of the reaction products. The temperatures reached are thus high enough to cause the gasification of phosphoric anhydride produced by the combustion. For this reason, P-nanothermites can be considered gas-generating materials.     

Synthesis and characterization of two quaternary thorium chalcophosphates: Cs4Th2P6S18 and Rb7Th2P6Se21

Two new thorium chalcophosphates have been synthesized by the reactive flux method and characterized by single-crystal X-ray diffraction, diffuse reflectance, and Raman spectroscopy: Cs4Th2P6S18 (I); Rb7Th2P6Se21 (II). Compound I crystallizes as colorless blocks in the triclinic space group P1 (No. 2) with a = 12.303(4) A, b = 12.471(4) A, c = 12.541(4) A, alpha = 114.607(8) degrees, beta = 102.547(6) degrees, gamma = 99.889(7) degrees, and Z = 2. The structure consists of (Th2P6S18)(4-) layers separated by layers of cesium cations and only contains the (P2S6)(4-) building block. Compound II crystallizes as red blocks in the triclinic space group P1 (No. 2) with a = 11.531(3) A, b = 12.359(4) A, c = 16.161(5) A, alpha = 87.289(6) degrees, beta = 75.903(6) degrees, gamma = 88.041(6) degrees, and Z = 2. The structure consists of linear chains of (Th2P6Se21)(7-) separated by rubidium cations. Compound II contains both the (PSe4)(3-) and (P2Se6)(4-) building blocks. Both structures may be derived from two known rare earth structures where a rare earth site is replaced by an alkali or actinide metal to form these novel structures. Optical band gap measurements show that compound I has a band gap of 2.8 eV and compound II has a band gap of 2.0 eV. Solid-state Raman spectroscopy of compound I shows the vibrations expected for the (P2S6)(4-) unit. Raman spectroscopy of compound II shows the vibrations expected for both (PSe4)(3-) and (P2Se6)(4-) units. Our work shows the remarkable diversity of the actinide chalcophosphate system and demonstrates the phase space is still ripe to discover new structures.     

Reactivity of rhenium(V) oxo Schiff base complexes with phosphine ligands: rearrangement and reduction reactions

The symmetric rhenium(V) oxo Schiff base complexes trans-[ReO(OH2)(acac2en)]Cl and trans-[ReOCl(acac2pn)], where acac2en and acac2pn are the tetradentate Schiff base ligands N,N'-ethylenebis(acetylacetone) diimine and N,N'-propylenebis(acetylacetone) diimine, respectively, were reacted with monodentate phosphine ligands to yield one of two unique cationic phosphine complexes depending on the ligand backbone length (en vs pn) and the identity of the phosphine ligand. Reduction of the Re(V) oxo core to Re(III) resulted on reaction of trans-[ReO(OH2)(acac2en)]Cl with triphenylphosphine or diethylphenylphosphine to yield a single reduced, disubstituted product of the general type trans-[Re(III)(PR3)2(acac2en)]+. Rather unexpectedly, a similar reaction with the stronger reducing agent triethylphosphine yielded the intramolecularly rearranged, asymmetric cis-[Re(V)O(PEt3)(acac2en)]+ complex. Reactions of trans-[Re(V)O(acac2pn)Cl] with the same phosphine ligands yielded only the rearranged asymmetric cis-[Re(V)O(PR3)(acac2pn)]+ complexes in quantitative yield. The compounds were characterized using standard spectroscopic methods, elemental analyses, cyclic voltammetry, and single-crystal X-ray diffraction. The crystallographic data for the structures reported are as follows: trans-[Re(III)(PPh3)2(acac2en)]PF6 (H48C48N2O2P2Re.PF6), 1, triclinic (P), a = 18.8261(12) A, b = 16.2517(10) A, c = 15.4556(10) A, alpha = 95.522(1) degrees , beta = 97.130(1) degrees , gamma = 91.350(1) degrees , V = 4667.4(5) A(3), Z = 4; trans-[Re(III)(PEt2Ph)2(acac2en)]PF6 (H48C32N2O2P2Re.PF6), 2, orthorhombic (Pccn), a = 10.4753(6) A, b =18.4315(10) A, c = 18.9245(11) A, V = 3653.9(4) A3, Z = 4; cis-[Re(V)O(PEt3)(acac2en)]PF6 (H33C18N2O3PRe.1.25PF6, 3, monoclinic (C2/c), a = 39.8194(15) A, b = 13.6187(5) A, c = 20.1777(8) A, beta = 107.7730(10) degrees , V = 10419.9(7) A3, Z = 16; cis-[Re(V)O(PPh3)(acac2pn)]PF6 (H35C31N2O3PRe.PF6), 4, triclinic (P), a = 10.3094(10) A, b =12.1196(12) A, c = 14.8146(15) A, alpha = 105.939(2) degrees , beta = 105.383(2) degrees , gamma = 93.525(2) degrees , V = 1698.0(3) A3, Z = 2; cis-[Re(V)O(PEt2Ph)(acac2pn)]PF6 (H35C23N2O3PRe.PF6), 5, monoclinic (P2(1)/n), a = 18.1183(18) A, b = 11.580(1) A, c = 28.519(3) A, beta = 101.861(2) degrees , V = 5855.9(10) A(3), Z = 4.