Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

Effect of Substituents and of Wavelength of Irradiation on the Photo-Fries-Rearrangement of Enol-Esters Preliminary Communication

Enol-esters 1a-1e undergo clean Photo-Fries-rearrangements without side reactions. With anthroyl derivatives the reaction is observed only at 254 nm, not at 366 nm.     

Elastic and Raman scattering of 8.5–11.4 MeV photons from 159Tb, 165Ho and 237Np

Differential cross sections for elastic and inelastic Raman scattering from the deformed heavy nuclei ¹⁵⁹Tb, ¹⁶⁵Ho and ²³⁷Np were measured at five energies between 8.5 and 11.4 MeV. Angular distributions at four angles between 90° and 140° for both elastic and inelastic scattering at 9.0 and 11.4 MeV were also measured. The monoenergetic photons were obtained from thermal neutron capture in Ni and Cr. All the angular distributions and the elastic and Raman scattering at the higher energies are in good overall agreement with theoretical predictions. The theory is based on a modified simple rotator model of the giant dipole resonance in which the effect of Delbrück scattering was included. A trend of both the elastic and Raman scattering at lower energies to be stronger than expected are suggested by the data. However, the ratio between the Raman and elastic scattering seem to be in good agreement with theory throughout the whole energy range. This shows that there is no need to introduce a direct nonresonant component to the imaginary part of the elastic scattering amplitude to explain the experimental data.     

Numerical calculations of Delbrück scattering amplitudes

The computation of the imaginary Delbrück amplitudes with the formulae of Constantini et al. gave results whose accuracy is greater by a factor of 10 than that available in the literature. The real amplitudes were computed using the code of Papatzacos and Mork. This computation was extended to all angular and energy regions of interest, and hence may be very useful in the interpretation of scattering experiments.     

Time-dependent-thermal-lensing (TDTL) studies on gas-phase azulene

Several aspects of the time-dependent-thermal-lensing (TDTL) technique were explored experimentally, including pressure effects, risetime behavior and falltime behavior. The experiments were performed using azulene vapor excited to high vibrational energies (≈ 17000 cm^?1). It was found that the energy appears in translational degrees of freedom at very early times, but the intrinsic limitations of the technique permitted only an estimate of the V-T rate constant for excited azulene deactivation by krypton. The result is discussed in terms of other recent experiments, and it is tentatively concluded that deactivation of excited azulene by unexcited azulene molecules may proceed primarily by a V-V process.     

Reductive Bergman-type cyclizations of cross-conjugated enediynes to fulvene and fulvalene anions: the role of the substituent

Various cross-conjugated enediynes undergo "Bergman-type" cycloaromatizations upon reduction with potassium metal, generating anions of fulvenes and fulvalene derivatives. This new anionic cyclization is considerably more facile than the classic Bergman cyclization with linear enediynes, creating highly reactive diradicals at -78 degrees C. Not all cross-conjugated enediynes yield cyclized dianions upon reduction; some give uncyclized, Y-shaped, cross-conjugated dianions, while others apparently yield radical-anions that either dimerize or persist as monomers. One system yields both a cyclized and an uncyclized dianion. The substituents are thus shown to be a critical factor in determining the outcome of the reduction. Cyclization occurs within a specific "window of opportunity" that is governed by the substituents.     

The chemistry of polycyclic arene imines. VII. Reactions of phenanthrene 9,10-imine with aromatic carboxaldehydes,carboxylic acids and acetylenic esters

The reactions of phenanthrene 9,10-imine ( 1 ) with aromatic aldehydes, benzoic acids and acetylenedi-carboxylic esters were investigated. The aldehydes were shown to give 1-[N-(arylmethylidene)-9-phenanthreneamine-10-yl]-1a,9b-dihydrophenanthro[9,10-b]azirine 2. The ‘dimeric’ structure of these products was established by X-ray diffraction analysis. The carboxylic acids proved to form in the presence of dicyclohexylcarbodiimide, N-aroylphenanthrene 9,10-imines 7 , that readily undergo rearrangement to N-aroyl-9-phenanthrenamines 8. Esters of acetylenedicarboxylic acid gave the corresponding esters of (Z)-2-(1a,9b-dihydrophenanthro[9,10-b]azirine-1-yl)-2-butendioic acid 10 .     

The reaction of phenylglyoxal with 2-aminoalcohols. Rearrangement of 2-acyloxazolidines to 2-hydroxy-5,6-dihydro-1,4-oxazines

The reactions of phenylglyoxal hydrate with (R)-phenylglycinol and with (1S,2R)-norephedrine were investigated. The expected 2-benzoyloxazolidines 3 and 2a , respectively are initially formed, and undergo a fast, spontaneous stereospecific rearrangement to the corresponding 2-hydroxy-3-phenyl-5,6-dihydro-1,4-oxazines 4 and 5 respectively. The mechanism of this new rearrangement is discussed. The structures and stereochemistry of 4 and 5 were established by X-ray diffraction analysis.