Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

Practical syntheses of [1,3,5]‐triazine dendritic molecules on solid supports

A practical and divergent synthesis of supported [1,3,5]‐triazine dendritic molecules on Wang resin, PEGA resin, SynPhase? Lanterns, and silica gel is described. The alkylamine linkers used allow derivatization with functionality for both synthetic (e.g., supported reagent and scavenger activity) and chemical biology applications. The use of supported intermediates allows differentiation of symmetric linkers without the need for protecting group chemistry. The synthetic route uses inexpensive, readily available starting materials in a straightforward and scaleable strategy. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2248–2259, 2006     

Low temperature specific heat calculations for amorphous polystyrene

Numerous attempts have previously been made to explain low temperature specific heat anomalies in glasses. The potential value of low frequency Raman data in interpreting such results is demonstrated here: the density of states for the 60 cm^?1 side-group mode in amorphous polystyrene is calculated from Raman measurements. Using the harmonic approximation, the contribution to the specific heat from this side-group motion is calculated and added to the Debye term. The form of C/T³ between 1.5 and 4 K is shown to be reproduced, although the magnitude of C/T³ is only about 70% of experimental values. This discrepancy and the anomalous behaviour below 1.5 K is believed to the due to lower-lying vibrational states of the system, perhaps not accessible through Raman scattering experiments.     

Stereoselective reductase-catalysed deoxygenation of sulfoxides in aerobic and anaerobic bacteria

Direct and indirect evidence, of unexpected stereoselective reductase-catalysed deoxygenations of sulfoxides, was found. The deoxygenations proceeded simultaneously, with the expected dioxygenase-catalysed asymmetric sulfoxidation of sulfides, during some biotransformations with the aerobic bacterium Pseudomonas putida UV4. Stereoselective reductase-catalysed asymmetric deoxygenation of racemic alkylaryl, dialkyl and phenolic sulfoxides was observed, without evidence of the reverse sulfoxidation reaction, using anaerobic bacterial strains. A purified dimethyl sulfoxide reductase, obtained from the intact cells of the anaerobic bacterium Citrobacter braakii DMSO 11, yielded, from the corresponding racemates, enantiopure alkylaryl sulfoxide and thiosulfinate samples.     

An investigation of mononuclear and binuclear palladium(II) complexes of egta (egta = glycine,N,N′-(1,2-ethanediylbis(oxy-2,1-ethanediyl)bis[N-carboxymethyl]) by 1H-, 13C- and 15N-n.m.r. methods

1:1 and 2:1 palladium(II) complexes of egta^4– (egta^4– = glycine, N,N-(1,2-ethanediylbis)(oxy-2,1-ethanediyl)bis[N-carboxymethyl]) were prepared by 1:1 and 2:1 addition of K₂PdCl₄ to K₄egta, and examined by ¹H-, ¹³C- and ¹⁵N-n.m.r. methods. The 1:1 complex, [Pd(egta)]^2– in solution, utilizes a square-planar coordination comprised of two nitrogen and two glycinato carboxylate donors of egta^4–, leaving two glycinato carboxylates pendant. The complex has a cis-(R,S) stereochemistry which places both pendant carboxylates below the PdN₂O₂ square plane and the tether backbone of egta^4– in the up, up sense above the same plane. The cis-(R,S) assignment was assisted by computer simulations of the ¹³C-n.m.r. spectrum for four possible isomers. Only cis-(R,S) and trans-(R,R) calculated ¹³C-spectra were compatible with the observed ¹³C-n.m.r. pattern. The HH NOESY spectrum of [Pd(egta)]^2– detects long range coupling of the backbone –OCH₂CH₂O– linkage with both coordinated and pendant glycinato CH₂ moieties. The cis-(R,S) isomer's tortional movements allow such contacts whereas a trans-(R,R) isomer does not. The 2:1 complex, [Pd₂(egta)(H₂O)₂] in solution has an extended-chain structure with each palladium(II) center coordinated in the mer-iminodiacetate-like coordination with two bound glycinato-functionalities.     

Near-perfect matrices

A 0, 1 matrixA isnear-perfect if the integer hull of the polyhedron {x0: Ax } can be obtained by adding one extra (rank) constraint. We show that in general, such matrices arise as the cliquenode incidence matrices of graphs. We give a colouring-like characterization of the corresponding class of near-perfect graphs which shows that one need only check integrality of a certain linear program for each 0, 1, 2-valued objective function. This in contrast with perfect matrices where it is sufficient to check 0, 1-valued objective functions. We also make the following conjecture: a graph is near-perfect if and only if sequentially lifting any rank inequality associated with a minimally imperfect graph results in the rank inequality for the whole graph. We show that the conjecture is implied by the Strong Perfect Graph Conjecture. (It is also shown to hold for graphs with no stable set of size eleven.) Our results are used to strengthen (and give a new proof of) a theorem of Padberg. This results in a new characterization of minimally imperfect graphs: a graph is minimally imperfect if and only if both the graph and its complement are near-perfect.The research has partially been done when the author visited Mathematic Centrum, CWI, Amsterdam, The Netherlands.     

Five coordination modes of 4-aminopyrimidine with N-hydroxy-ethylethylenediaminetriacetatoruthenate(II)

Summary The complex [Ru^II(hedta)(4NH₂pym)]⁻, hedta³⁻ = N-hydroxyethylethylenediaminetriacetate, 4NH₂pym = 4-aminopyrimidine, exists at pH 7 as five different coordination isomers, which are most readily distinguished by their electrochemical waves in comparison with the 2-aminopyridine (2NH₂py) complex. The 2NH₂py complex exhibits N(1) (pyridine bound), exo-NH₂ (amine bound) and N(1), NH₂-chelated species. The 4NH₂pym complex forms N(1), exo-amine and N(3), NH₂-chelated isomers analogues to the 2NH₂py species, but also engages in η² (olefin bound) coordination of the dearomatized 4NH₂pym ring in C(5)–C(6), and another η² type of complex involving electron density between N(1) and N(3) of the ring (η³ form). N(1), η² and η³ isomers have also been detected for unsubstituted pyrimidine (pym), 4-methylprimidine (4CH₃pym) and 2-aminopyrimidine (2NH₂pym). Electrochemical waves (V versus NHE) for the five isomers are assigned as follows: (Ru^II/III) exo-NH₂ (0.06 V), N(1) (0.29 V), η² (0.49 V); (Ru^II/III) η³ (0.76 V); N(3), NH₂-chelated (1.09 V).     

The influence of pentaamminerhodium(III) on the1H n.m.r. spectra and pyrrole pKas of coordinated imidazoles and pyrazoles

Summary The following [(NH₃)₅RhLH]Cl₃ salts were preparedvia the [(NH₃)₅Rh(O₃SCF₃)](O₃SCF₃)₂ synthetic route; LH=1-methylimidazole (1CH₃imH), 2-methylimidazole (2CH₃imH), 4-methylimidazole (4CH₃imH), 5-methylimidazole(5CH₃imH), and pyrazole (pyzH). pK_a's at 25.0°C were determined for [(NH₃)₅RhLH]³⁺ complexes as follows: 2CH₃imH, 10.4±0.1; 5CH₃imH/4CH₃imH isomer mixture, 10.3±0.1; pyzH, 6.54±0.05. The influence on the pK_a's of imidazoles is dominated by withdrawal of the rhodium(III) centre and may be compensated by the presence of ring methylation by only 0.5log units for cobalt(III) and rhodium(III) derivatives, compared to 1.3 units for the -withdrawing ruthenium(III) centre. In the case of the -acceptor pyrazole ring, [(NH₃)₅Rh]³⁺ is observed to serve as a slight -donor and raises the pK_a above the cobalt(III) analogue. The¹H n.m.r. spectra of [(NH₃)₅RhLH]³⁺ complexes of the substituted imidazoles and pyrazole exhibit a deshielding order. C–2H>C–5H>C–4H for imidazoles and C–3H>C–5H>C–4H for pyrazole, as do their cobalt(III) analogues. The magnitude of values (=_{free L}-_complex) are virtually the same as in the cobalt(III) systems which shows that TIP influences are unimportant compared to ring rehybridization in estabilishing chemical shifts for both the cobalt(III) and rhodium(III) complexes. The imidazolato and pyrazolato complexes exhibit resonances upfield of the respective substituted imidazole or pyrazole complex in keeping with more negative charge on the rings; the influence is largest at C–2H of imidazolates and C–3H of pyrazolate.     

Diastereoselective synthesis of cyclopropane amino acids using diazo compounds generated in situ

A simple and high-yielding method for the preparation of cyclopropane amino acids is described. The novel method involves the one-pot cyclopropanation of readily available dehydroamino acids using aryl and unsaturated diazo compounds generated in situ from the corresponding tosylhydrazone salts. It was found that thermal 1,3-dipolar cycloaddition followed by nitrogen extrusion gave the cyclopropane amino acid derivatives with good E selectivity, while reactions in the presence of meso-tetraphenylporphyrin iron chloride gave predominantly the corresponding Z isomers. The synthetic utility of this process was demonstrated in the synthesis of (+/-)-(Z)-2,3-methanophenylalanine [(+/-)-(Z)-1], the anti-Parkinson (+/-)-(E)-2,3-methano-m-tyrosine [(+/-)-(E)-2], and the natural product (+/-)-coronamic acid [(+/-)-3].