A generalization of perfect graphs—i-perfect graphs

Let i be a positive integer. We generalize the chromatic number _X(G) of G and the clique number o(G) of G as follows: The i-chromatic number of G, denoted by _X(G), is the least number k for which G has a vertex partition V₁, V₂,…, V_k such that the clique number of the subgraph induced by each V_j, 1 ≤ j ≤ k, is at most i. The i-clique number, denoted by o_i(G), is the i-chromatic number of a largest clique in G, which equals [o(G/i]. Clearly _X1(G) = _X(G) and o₁(G) = o(G). An induced subgraph G′ of G is an i-transversal iff o(G′) = i and o(G − G′) = o(G) − i. We generalize the notion of perfect graphs as follows: (1) A graph G is i-perfect iff _Xi(H) = o_i(H) for every induced subgraph H of G. (2) A graph G is perfectly i-transversable iff either o(G) ≤ i or every induced subgraph H of G with o(H) > i contains an i-transversal of H. We study the relationships among i-perfect graphs and perfectly i-transversable graphs. In particular, we show that 1-perfect graphs and perfectly 1-transversable graphs both coincide with perfect graphs, and that perfectly i-transversable graphs form a strict subset of i-perfect graphs for every i ≥ 2. We also show that all planar graphs are i-perfect for every i ≥ 2 and perfectly i-transversable for every i ≥ 3; the latter implies a new proof that planar graphs satisfy the strong perfect graph conjecture. We prove that line graphs of all triangle-free graphs are 2-perfect. Furthermore, we prove for each i greater than or equal to2, that the recognition of i-perfect graphs and the recognition of perfectly i-transversable graphs are intractable and not likely to be in co-NP. We also discuss several issues related to the strong perfect graph conjecture. © 1996 John Wiley & Sons, Inc.     

Chiral Nematic Cellulose Acetate Films; Reflectivity and Effect of Deacetylation

Cellulose acetate samples with a range of degrees of substitution (DS) were prepared by deacetylation of cellulose acetate (DS = 2.5). Chiral nematic solutions of the samples were prepared in formic acid. A handedness reversal from left to right was observed as the DS used to prepare the mesophase increased from 1.8 to 2.4. By selection of a suitably long-pitch mesophase, chiral nematic films cast from a CA (DS = 2.2)/formic acid solution showed a feint reflection of circularly polarized blue light. Deacetylation of this CA film gave a chiral nematic cellulose film with a more intense reflection band at the same wavelength. The improvement in intensity is attributed to the higher intrinsic birefringence of cellulose compared to cellulose acetate.     

Generation and trapping of monomeric metaphosphoric acid esters in solution: Mechanistic investigation into the fragmentation of mixed carbonic-phosphoric acid anhydrides and the chelotropic breakdown of cyclic pyrocarbonate esters

The generation of alkyl-substituted monomeric metaphosphoric acid esters (ROPO₂) 4 in solution is described, utilizing two different methods based on the condensation of alkyl phosphorodichloridates either with anhydrous potassium hydrogen carbonate or with the novel disodium pyrocarbonate salt 5, which is formed in quantitative yield from diethyl pyrocarbonate by treatment with two equivalents of sodium trimethylsilanolate in THF at 0°C. The first method is believed to proceed via a transient mixed carbonic-phosphoric anhydride 3, which decomposes at room temperature with release of 1 mol each of hydrogen chloride and carbon dioxide to produce 4. In the other method, ³¹P NMR spectroscopy indicated the involvement of a cyclic pyrocarbonate phosphate (2-alkoxy-1,3,5-trioxaphosphorinane-4,6-dione-2-oxide) 6, which decomposed in situ with the release of 2 mol of carbon dioxide. In both cases, the metaphosphate thus formed spontaneously self-condensed to produce polymeric species with P-O-P bonds having characteristic ³¹P NMR signals clustered in the δ -12 and -24 regions. When metaphosphate 4 is generated by either process in the presence of styrene oxide, polymerization is avoided, and trapping occurs with the exclusive formation of a diastereomeric mixture of 2-alkoxy-1,3,2-dioxaphospholane-2-oxides 7 (cis, trans). In order to shed some mechanistic light on the formation of 7, ³¹P NMR studies have been carried out on the individual trapping reactions between (—)-methyl metaphosphate 4d and racemic styrene oxide, as well as its optically pure forms. Based on evidence that these transformations proceed by inversion of configuration at the phenyl-bearing carbon of styrene oxide, a mechanism is invoked whereby the dipolar metaphosphate 4d reacts both as an electrophile and as a nucleophile in a self-catalyzed process via the activated complex 9d. © 1996 John Wiley & Sons, Inc.     

Morphological biosignatures from relict fossilised sedimentary geological specimens: a Raman spectroscopic study

Morphological biosignatures (features related to life) and associated terrestrial sedimentary structures that provide possible sampling targets for the remote astrobiological exploration of planets have been analysed using Raman spectroscopic techniques. The spectral data from a suite of samples comprising crypto‐chasmoendoliths, preserved microbial filaments and relict sedimentary structures comprise a preliminary database for the establishment of key Raman biosignatures. This will form the basis for the evaluation of prototype miniaturised instrumentation for the proposed ESA ExoMars mission scheduled for 2013. The Raman spectral biosignatures of carotenoids and scytonemin, organic biomolecules characteristic of the cyanobacterial colonisation of geological matrices and biogeologically modified minerals are also identifiable in the sedimentary specimen materials. The results of this study demonstrate the basis of the molecular recognition of extinct and extant exobiology that will feed into the elemental structural analyses of morphological structures provided by associated SEM, XRD and laser‐induced breakdown spectroscopy (LIBS) techniques on robotic analytical landers. Copyright © 2007 John Wiley & Sons, Ltd.     

Progress in miniature liquid film combustors: Double chamber and central porous fuel inlet designs

One of the key limits to miniaturizing the size of liquid fueled combustors is the atomization process applied in meso-scale systems. A single-wall fuel-film combustor was introduced recently as one of the successful liquid fuel combustors at the meso-scale. Instead of atomizing the fuel, film combustors spread out a liquid film along the wall and absorb the heat transferred from the flame for vaporization. With a single-wall film design, however, there are some unexpected and disadvantageous combustion phenomena. This paper attempts to improve the single-wall film combustor by exploring separately a double chamber concept and a central porous fuel delivery concept. These two configurations help describe the limits and the potential of liquid fuel-film miniature combustors. The double chamber design demonstrates how heat transfer issues can be overcome by injecting the fuel-film on the outside of the primary combustor wall rather than on the inside, and the second design demonstrates a flame-holding mechanism using a porous material set on the bottom of the chamber. The combustion behavior in these two configurations is compared with that in the original single-wall miniature fuel-film combustor, revealing new aspects that are relevant to portable power generation with high specific energy and power.     

The arene oxede-oxepine valence isomerization; a dynamic nmr investigation using a prochiral substituent

The geminal -methytene protons in 3-ethylnaphthalene-l,2-oxide show chemical shift nonequivalence in ¹H NMR spectra recorded at ambient temperature consistent with a high barrier to enantiomerization via the orepine tautomer. The methylene protons in -benzoyloxytoluene-2,3-oxide which are a singlet at ambient temperature split into an AB system at -135 °C and there is evidence of ca. 5% of the oxepine isomer. Lineshape analysis of the coalescing arene oxide AB Signals provides the activation parameters for the valence isomerization/degenerate racemization process: ΔGXXX = 7.6 kcal mol^-1, ΔHXXX = 8.6 kcal mol^-l, ΔSXXX= 5.8 cal K^-l mol^-1.