Acyclic edge coloring of graphs with maximum degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a′(G)?Δ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)?4, with the additional restriction that m?2n?1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m?2n, when Δ(G)?4. It follows that for any graph G if Δ(G)?4, then a′(G)?7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009     

Elastic scattering of electrons by benzene

Relative differential cross-sections for the elastic scattering of electrons from benzene have been measured at incident energies 300, 500, 700 and 900eV and for scattering angles between 30° and 120°. The results are discussed and compared with the independent atom model (IAM) calculations. Two different sets of scattering amplitudes for the constituent atoms of benzene were used in these calculations, one obtained from first Born approximation and the other from partial wave analysis of the Dirac equation. Only the static interaction was taken into account in the calculations. The higher the incident energy, the better is the observed agreement between experiment and theory. This indicates that at higher energies, absorption, exchange and polarization effects are not significant as compared to the static interaction and that the IAM satisfactorily predicts the interference of scattering from the individual atoms of C₆H₆.     

Stable Crystalline Forms of Na Polysulfides: Experiment versus Ab Initio Computational Prediction

For the design of light‐metal–sulfur batteries and for the understanding of their performance, knowledge on the stable crystalline polysulfides is very important. We confronted experimental and ab initio crystal structure prediction studies on the stability of Na polysulfides. The selected evolutionary‐based structure‐prediction algorithm was able to quickly and correctly predict the thermodynamically stable crystalline forms of Na polysulfides with small unit cells. For Na polysulfides with large unit cells, the algorithm correctly proposed short unbranched polysulfide chains to be energetically favorite structural motifs, but could not find proper three‐dimensional structures in the limited number of generations. Experimentally, the polysulfides were studied by X‐ray diffraction and ²³Na solid‐state NMR spectroscopy. Complemented by calculations of the isotropic chemical shifts and quadrupolar coupling constants, NMR spectroscopy proved to be an excellent tool for the examination of Na polysulfides, because it allowed easy distinction and quantification of components in the samples.     

Optical sectioning in reciprocal fiber-optic based chromatic confocal microscope

A theoretical analysis for studying the optical sectioning in a chromatic confocal microscope employing the same single mode fiber for illumination and detection is described. The confocality achieved by different wavelengths due to the fixed size of the fiber core is investigated using diffraction theory. The detected signal level for the different wavelengths is also calculated. The ratio of the numerical aperture of the lens and the fiber is found to play a strong role in the performance of the system.     

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) ≤ 2r(G) + c, where r(G) denotes the radius of G and \({c \in \{0, 1, 2\}}\) . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius [1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time \({(2 + \frac{2}{r(G)})}\) -factor approximation algorithms.     

Sequential determination of fat- and water-soluble vitamins in Rhodiola imbricata root from trans-Himalaya with rapid resolution liquid chromatography/tandem mass spectrometry

A rapid method was developed to determine both types of vitamins in Rhodiola imbricata root for the accurate quantification of free vitamin forms. Rapid resolution liquid chromatography/tandem mass spectrometry (RRLC–MS/MS) with electrospray ionization (ESI) source operating in multiple reactions monitoring (MRM) mode was optimized for the sequential analysis of nine water-soluble vitamins (B₁, B₂, two B₃ vitamins, B₅, B₆, B₇, B₉, and B₁₂) and six fat-soluble vitamins (A, E, D₂, D₃, K₁, and K₂). Both types of vitamins were separated by ion-suppression reversed-phase liquid chromatography with gradient elution within 30 min and detected in positive ion mode. Deviations in the intra- and inter-day precision were always below 0.6% and 0.3% for recoveries and retention time. Intra- and inter-day relative standard deviation (RSD) values of retention time for water- and fat-soluble vitamin were ranged between 0.02–0.20% and 0.01–0.15%, respectively. The mean recoveries were ranged between 88.95 and 107.07%. Sensitivity and specificity of this method allowed the limits of detection (LOD) and limits of quantitation (LOQ) of the analytes at ppb levels. The linear range was achieved for fat- and water-soluble vitamins at 100–1000 ppb and 10–100 ppb. Vitamin B-complex and vitamin E were detected as the principle vitamins in the root of this adaptogen which would be of great interest to develop novel foods from the Indian trans-Himalaya.     

‘Counter-intuitive’ regioselectivity, subtle steric and solvation effects in lithiation of cyclic tertiary aralkylamines

Lithiation of a series of cyclic aralkyl tertiary amines with sec-BuLi in various solvents has been studied. There is a subtle sensitivity to steric factors and lithium coordinating solvents/additives have an adverse effect. ortho-Lithiation is observed only in the case of an eight-membered cyclic amine and the ease of benzylic lithiation with respect to nitrogen is in the surprising order γ > β ? α, δ. These observations are discussed in the context of nitrogen coordination promoted lithiation.     

Adducts of Bicyclic (Alkyl)(Amino)Carbene with ECl3 and Three Electron Reduction Thereof: Syntheses of BICAAC Stabilized E−E Bonded Compounds (E=P,Sb)

The stoichiometric reaction of bicyclic (alkyl)(amino)carbene (BICAAC) with group 15 chlorides, ECl₃ (E=P, Sb) to form the Lewis adducts BICAAC:ECl₃ (E=P ( 1 ), Sb ( 2 )) has been investigated in the present work. The BICAAC smoothly reacts with PPhCl₂ to form [BICAAC:PPhCl₂] which on in-situ two electron reduction with 2 equivalents of KC₈ afforded the phosphinidene complex, BICAAC=P-Ph ( 3 ). Complete dechlorination of the BICAAC-ECl₃ adducts 1 and 2 with 3 equivalents of KC₈ leads to three-electron reduction to afford low-valent trans bent bis(BICAAC)E₂ complexes (E=P ( 4 ) and Sb ( 5 )). These complexes are the first examples of BICAAC adducts with pnictogens and all the complexes have been characterized by different spectroscopic techniques and their solid-state structure have been elucidated by single crystal X-ray diffraction.     

Selective solar photocatalytic oxidation of ethylbenzene on C,N, and S doped TiO<Subscript>2</Subscript>

The liquid-phase photo-oxidation of ethylbenzene (EB) is investigated in solar light with air/O₂/N₂ at atmospheric pressure, in a batch reactor using acetonitrile medium. It is carried out over TiO₂ doped with C, N, and S (TCNS) photocatalyst samples. The photocatalytic oxidation yielded acetophenone (33%) and 1-phenylethanol (21%) at 56% conversion of EB during a 6-h irradiation time. This product distribution indicates that C–H bond activation has occurred only at the alkyl chain. The effects of the EB to water content ratio, amount of photocatalyst and its sustainability, pH, have been studied. It is illustrated that the reaction carried out by this environmentally friendly photocatalysis is truly heterogeneous under mild conditions using solar light and no waste generation. An optimum loading of TCNS5 was observed for the photo-oxidation of EB.