On products of graphs and regular groups

A graphX is called a graphical regular representation (GRR) of a groupG if the automorphism group ofX is regular and isomorphic toG. Watkins and Nowitz have shown that the direct productG×H of two finite groupsG andH has aGRR if both factors have aGRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complementX′ of a finite or infinite graphX is prime with respect to cartesian multiplication ifX is composite and not one of six exceptional graphs.     

Tiled partial cubes

In the quest to better understand the connection between median graphs, triangle‐free graphs and partial cubes, a hierarchy of subclasses of partial cubes has been introduced. In this article, we study the role of tiled partial cubes in this scheme. For instance, we prove that almost‐median graphs are tiled and that tiled partial cubes are semi‐median. We also describe median graphs as tiled partial cubes without convex Q and extend an inequality for median graphs to a larger subclass of partial cubes. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 91–103, 2002     

Stabilization of parallel triplexes by twisted intercalating nucleic acids (TINAs) incorporating 1,2,3-triazole units and prepared by microwave-accelerated click chemistry

A highly efficient method for postsynthetic modification of unprotected oligonucleotides incorporating internal insertions of (R)-1-O-(4-ethynylbenzyl)glycerol has been developed through the application of click chemistry with water-insoluble pyren-1-yl azide and water-soluble benzyl azide and acceleration by microwave irradiation. The twisted intercalating nucleic acids (TINAs) obtained in these reactions, possessing bulged insertions of (R)-3-O-{4-[1-(pyren-1-yl)-1H-1,2,3-triazol-4-yl]benzyl}glycerol (7), formed parallel triplexes with thermal stabilities of 20.0, 34.0, and 40.0 degrees C at pH 7.2 in the cases of one, two, or three insertions of 7, respectively, separated by three nucleic bases. An oligonucleotide with four insertions of 7--each between three nucleic bases in the sequence--was unable to form complexes with complementary single- or double-stranded DNAs, as a result of self-aggregation of the pyrene moieties. This assumption was supported by the formation of a very strong excimer band at 460 nm in the fluorescence spectra. Molecular modeling of the parallel triplex with bulged insertion of the monomer 7 in the triplex-forming oligonucleotide (TFO) showed that only the pyrene moiety was stacking between the bases of the dsDNA, whereas 1,2,3-triazole did not participate in the triplex stabilization. Thermal denaturation studies of the duplexes and triplexes, as well as the fluorescence properties of TINA-triazole 7, are discussed and compared with previous studies on TINA.     

General Lower Bounds for the Minor Crossing Number of Graphs

There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor crossing number to string graphs and establish a lower bound for the standard crossing number in terms of Randič index.     

Planar Crossing Numbers of Graphs of Bounded Genus

Pach and Tóth proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c ^g dn, for a constant c>1. We improve on this result by decreasing the bound to O(dgn), and also prove that our result is tight within a constant factor. Our proof is constructive and yields an algorithm with time complexity O(dgn). As a consequence of our main result, we show a relation between the planar crossing number and the surface crossing number.     

Prediction by 13C NMR of regioselectivity in 1,3‐dipolar cycloadditions of acridin‐9‐yl dipolarophiles

Strong correlation was found between ¹³C NMR chemical shifts of dipolarophilic CH?CH carbons and regioselectivity in 1,3‐dipolar cycloadditions of new acridin‐9‐yl dipolarophiles with stable benzonitrile oxides (BNO). Accordingly, two starting dipolarophiles, (acridin‐9‐yl)‐CH?CH‐R (R = COOCH₃ or Ph), reacted with three BNOs (2,4,6‐trimethoxy, 2,4,6‐trimethyl, and 2,6‐dichloro) to give a mixture of two target isoxazoline regioisomers in which the acridine was bound either to isoxazoline C‐4 carbon (4‐Acr) or C‐5 one (5‐Acr). Methyl 3‐(acridin‐9‐yl)propenoate afforded major 4‐(acridin‐9‐yl)‐isoxazoline‐5‐carboxylates (4‐Acr) and minor 5‐(acridin‐9‐yl)‐4‐carboxylates (5‐Acr). 9‐(2‐Styryl)acridine regiospecifically afforded only 4‐Acr cycloadducts. The ratios of regioisomers were compared with analogous reactions of acridin‐4‐yl dipolarophiles. Regioselectivity was dependent on a polarity of the CH?CH bond, donor effects in BNO, and stabilization by stacking of aromatic substituents in the products. Copyright © 2015 John Wiley & Sons, Ltd.     

Strong deshielding in aromatic isoxazolines

Very strong proton deshielding was found in di/tri‐aromatic isoxazoline regioisomers prepared from acridin‐4‐yl dipolarophiles and stable benzonitrile oxides (BNO). Three alkenes, (acridin‐4‐yl)‐CH?CH‐R (R = COOCH₃, Ph, and CONH₂), reacted with three BNO dipoles (2,4,6‐trimethoxy, 2,4,6‐trimethyl, 2,6‐dichloro) to give pairs of target isoxazolines with acridine bound to C‐4 or C‐5 carbon of the isoxazoline (denoted as 4‐Acr or 5‐Acr). Regioselectivity was dependent on both the dipolarophile and dipole character. The ester and amide dipolarophile displayed variable regioselectivity in cycloadditions whereas the styrene one afforded prevailing 4‐Acr regioisomers. 2,4,6‐Trimethoxy‐BNO was most prone to form 5‐Acr isoxazolines while mesitonitrile oxide gave major 4‐Acr isoxazolines. Basic hydrolysis of the amide cycloadduct led to an unexpected isoxazolone product. The structure of the target compounds was studied by NMR, MS, and X‐ray crystallography. Copyright © 2015 John Wiley & Sons, Ltd.