A stability version for a theorem of Erdős on nonhamiltonian graphs

Let $n,d$ be integers with $1\le d\le ?\frac{n?1}{2}?$, and set $h(n,d)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)+d^2$ and $e(n,d)?max\{h(n,d),h(n,?\frac{n?1}{2}?)\}$. Because $h(n,d)$ is quadratic in $d$, there exists a $d_0\left(n\right)=(n∕6)+O\left(1\right)$ such that

$e(n,1)>e(n,2)>?>e(n,d_0)=e(n,d_0+1)=?=e\left(n,?\frac{n?1}{2}?\right).$

A theorem by Erd?s states that for $d\le ?\frac{n?1}{2}?$, any $n$-vertex nonhamiltonian graph $G$ with minimum degree $\delta \left(G\right)\ge d$ has at most $e(n,d)$ edges, and for $d>d_0\left(n\right)$ the unique sharpness example is simply the graph $K_n?E\left(K_{?(n+1)∕2?}\right)$. Erd?s also presented a sharpness example $H_{n,d}$ for each $1\le d\le d_0\left(n\right)$.We show that if $d<d_0\left(n\right)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $\delta \left(G\right)\ge d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d)?e(n,d+1)=n?3d?2\ge n∕2$ whenever $d<d_0\left(n\right)?1$.     

Spectroscopic Detection of Tetracationic Porphyrin H-Aggregation on Polyanionic Matrix of Inorganic Polyphosphate

Self-assembly of tetracationic porphyrin TMPyP⁴⁺ onto polyanionic matrix of inorganic polyphosphate (PPS) in aqueous solutions has been studied in a wide range of molar phosphate-to-dye ratios using techniques of polarized fluorescence, absorption, resonance Raman spectroscopy and static light scattering. The binding of TMPyP⁴⁺ to PPS is characterized by the binding constant of 3 × 10⁵ M⁻¹ and the cooperativity parameter of about 150. The fluorescence quenching of the bound TMPyP⁴⁺ evidences the stacking of the porphyrine chromophores. Under the stoichiometric binding ratio TMPyP⁴⁺ forms extended continuous face-to-face aggregates (so-called H-aggregates) which manifest themselves by a blue shift (12 nm) and a large hypochromisity (51%) of the Soret absorption band. Each face-to-face TMPyP⁴⁺ stack is formed with participation of four PPS chains. Formation of such columnar aggregates is promoted by the ability of PPS chains to take a helix conformation where negative charges are arranged along two oppositely situated rows with intercharge distance of 0.36 nm which corresponds to the thickness of the porphyrin π-electronic system. The ability of each PPS strand to be template for formation of two porphyrin stacks results in the integration of the adjacent stacks into higher-order aggregates which dimension was estimated from the fluorescence polarization data.     

One-pot regioselective vinylation of tetrazoles: preparation of 5-substituted 2-vinyl-2H-tetrazoles

The one-pot regioselective preparation of 5-aryl/alkyl-2-vinyl-2H-tetrazoles from 5-substituted tetrazoles via a very simple procedure using 1,2-dibromoethane and triethylamine without the need of any catalyst is described. The mechanism of this reaction is also discussed.     

Synthesis and properties of 4,5‐dihydrobenzo[e]imidazo[2,1‐c][1,4,2]diazaphosphinine derivatives

A number of 4,5‐dihydrobenzo[e]imidazo[2,1‐c][1,4,2]diazaphosphinine derivatives were prepared by the direct phosphorylation of 1‐(4‐Chlorophenylcarboxamido)‐2‐(1H‐1‐imidazolyl)‐5‐trifluoromethylbenzene in basic medium with phosphorus(III) bromide and dibromophenylphosphine. The tricyclic compounds 6a, 6b , and 9 having a trivalent phosphorus atom undergo the diazaphosphinine ring opening upon treatment with secondary amines in the presence of sulfur. © 2002 John Wiley & Sons, Inc. Heteroatom Chem 13:84–92, 2002; DOI 10.1002/hc.10000     

Mechanistic aspects of reactions of triphenylphosphine with electron‐deficient alkenes in acetic acid solution

Kinetic data provide evidence that proton transfer to the carbanion centre of a phosphonium zwitterion is the rate‐determining step in quarternization reactions of triphenylphosphine with electron‐deficient alkenes in acetic acid solution. This conclusion is based on the third‐order rate equation, the reactivity of the alkenes, and solvent isotope effects in deuteroacetic acid. Copyright © 2013 John Wiley & Sons, Ltd.     

Algebra of Screening Operators for the Deformed W n Algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W _n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W ₃ algebra. Received: 3 March 1997 / Accepted: 20 May 1997     

Steroid 19,B‐dinor‐8,10‐iso‐analogues

The Birch reduction of 3‐methoxy‐B‐nor‐8‐isoestra‐1,3,5(10)‐trienes followed by acid hydro­lysis produces steroid androgen 19,B‐dinor‐8,10‐iso‐analogues. By means of X‐ray analysis and correlation NMR spectroscopy of 16,16‐di­methyl‐D‐homo‐19,B‐dinor‐8‐isotestosterone, C₂₀H₃₀O₂, it is demonstrated that the main conformations in the crystal and in solution for two 19,B‐dinor‐8,10‐iso‐analogues are, in general, the same.     

Profiling of cyclic hexadepsipeptides roseotoxins synthesized in vitro and in vivo: a combined tandem mass spectrometry and quantum chemical study

High-performance liquid chromatography and tandem mass spectrometry (HPLC/MS/MS) was used for the detection of cyclic hexadepsipeptides roseotoxins produced by Trichothecium roseum. Roseotoxins were found in both submerged standard cultivation on CzapekDox medium and in vivo cultivation extract obtained from an apple. Roseotoxin chromatographic profiles from these two experiments were compared. Product-ion collision-induced dissociation (CID) spectra obtained on an ion trap (electrospray ionisation, ESI) were used for the identification of natural roseotoxins A, B, C and of minor destruxins A and B. The dissociation behavior of roseotoxins is discussed in terms of a fragmentation scheme proposed for describing the dissociation pathways of cyclic peptides. This scheme involves opening of the cyclopeptide ring via formation of oxazolone derivatives and fragmentation of the resulting linear species, which have a free N-terminus and an oxazolone ring at the C-terminus. Some aspects of this fragmentation scheme are underlined by modeling the dissociation channels of roseotoxin A using quantum chemical calculations. The structures of roseotoxin A and destruxin B were verified by nuclear magnetic resonance (NMR) spectroscopy. Structures of three new minor natural roseotoxins [Val(4)]RosA, [MeLxx(4)]RosA and [MeLxx(4)]RosB were deduced by ion cyclotron resonance Fourier transform mass spectrometry (ICR-FT-MS) and ion trap tandem mass spectrometry by examining the pre-separated roseotoxin fraction.     

Structures of cabergoline anhydrate form II and novel cabergoline solvates

Crystal forms of N-[3-(dimethylamino)propyl]-N-(ethylcarbamoyl)-6-allyl-ergoline-8β-carboxamide (cabergoline) originating from various solvents have been examined by X-ray diffraction at 298 or 150 K. Crystal structures of cabergoline anhydrate, (form II, P2₁2₁2₁) and solvates (all P2₁2₁2₁) with tert-butyl methyl ether (form VIII), cyclohexane (form XV), toluene (form IX), p-xylene (form XVI), and 1,2,4-trimethylbenzene (form XVII) are described. Conformation of cabergoline in these forms was compared with crystal structures of forms I and VII of cabergoline (P2₁) described in the literature. Despite a high degree of molecular conformational freedom, cabergoline possesses similar conformation in forms I, II, VIII, IX, XV, XVI, and XVII. Molecular conformations, crystal packing and the effect of the solvent on the former two properties are examined.