Filho H.B.M.,Federal University of Rio de Janeiro |
Machado R.C.S.,Inmetroinstituto Nacional Of Metrologia |
de Figueiredo C.M.H.,Federal University of Rio de Janeiro
Algorithmica | Year: 2016
The class of unichord-free graphs was recently investigated in the context of vertex-colouring (Trotignon and Vušković in J Graph Theory 63(1): 31–67, 2010), edge-colouring (Machado et al. in Theor Comput Sci 411(7–9): 1221–1234, 2010) and total-colouring (Machado and de Figueiredo in Discrete Appl Math 159(16): 1851–1864, 2011). Unichord-free graphs proved to have a rich structure that can be used to obtain interesting results with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies of colouring problems are found in subclasses of unichord-free graphs. In the present work, we investigate clique-colouring and biclique-colouring problems restricted to unichord-free graphs. We show that the clique-chromatic number of a unichord-free graph is at most 3, and that the 2-clique-colourable unichord-free graphs are precisely those that are perfect. Moreover, we describe an O(nm)-time algorithm that returns an optimal clique-colouring of a unichord-free graph input. We prove that the biclique-chromatic number of a unichord-free graph is either equal to or one greater than the size of a largest twin set. Moreover, we describe an (Formula presented.)-time algorithm that returns an optimal biclique-colouring of a unichord-free graph input. The clique-chromatic and the biclique-chromatic numbers are not monotone with respect to induced subgraphs. The biclique-chromatic number presents an extra unexpected difficulty, as it is not the maximum over the biconnected components, which we overcome by considering additionally the star-biclique-chromatic number. © 2016 Springer Science+Business Media New York
Sarpal A.S.,Inmetroinstituto Nacional Of Metrologia |
Silva S.R.,Inmetroinstituto Nacional Of Metrologia |
Silva P.R.M.,Inmetroinstituto Nacional Of Metrologia |
Monteiro T.V.,Inmetroinstituto Nacional Of Metrologia |
And 3 more authors.
Energy and Fuels | Year: 2015
A direct, rapid, and convenient method for the determination of iodine value (IV) of biodiesels from vegetable oils has been developed by the application of nuclear magnetic resonance (500 and 600 MHz 1H NMR) techniques. The method is based on the correlation of integrated intensities (Ius) of unsaturated hydrogen in the 1H NMR spectra of biodiesels with (1) the predetermined iodine values by the standard iodometry based method (EN 14111), (2) iodine values determined from the fatty acid profile by GC-MS method, and (3) iodine values from the absolute number of double bonds (ADB) determined by 1H NMR. The simple mathematical equation, IV = K(Ius) (K = 15.78), derived from the correlations, provides a direct method for the determination of iodine value in the range of 5-140 g of I2/(100 g). The constant K has been found to be independent of the nature and source of vegetable oils. It is equally applicable to blends of biodiesels, blends of biodiesel and oils in all proportions, and biodiesel samples containing glyceride impurities. The statistical comparison between NMR, EN 14111, and GC-MS methods indicates that NMR method provides equivalent results with acceptable precision. The quantitative NMR (q1H NMR) protocol for the optimization of acquisition and processing parameters have been established for achieving better precision on 500 and 600 MHz instruments. © 2015 American Chemical Society.