Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number eta(G) of a graph G is the largest integer n for which the complete graph K-n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, eta(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G square H of graphs. As the main result of this paper, we prove that eta(G(1) square G(2)) >= h root 1 (1 - o(1)) for any two graphs G(1) and G(2) with eta(G(1)) = h and eta(G(2)) = l. We show that the above lower bound is asymptotically best possible when h >= l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G(1) square G(2) square ... square G(k) be the ( unique) prime factorization of G. Then G satisfies Hadwiger's conjecture if k >= 2 log log chi(G) + c', where c' is a constant. This improves the 2 log chi(G) + 3 bound in [2] 2. Let G(1) and G(2) be two graphs such that chi(G1) >= chi(G2) >= clog(1.5)(chi(G(1))), where c is a constant. Then G1 square G2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G(d) (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan [2]. ( They had shown that the Hadiwger's conjecture is true for G(d) if d >= 3).

A quantum stochastic Lie-Trotter product formula

A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.

A total synthesis of the epoxyquinone based antifungal natural product (±)-ambuic acid

A total synthesis of the recently isolated polyketide natural product (+/-)-ambuic acid has been accomplished from the readily available Diels-Alder adduct of cyclopentadiene and 2-allyl-p-benzoquinone through a simple sequence with sound stereocontrol. (c) 2005 Elsevier Ltd. All rights reserved.

Note on the locally product and almost locally product structures

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X n of class C k . Every locally product space has certain almost locally product structures which transform the local tangent space to X n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.

Studies in oxasteroids—I : Synthesis of 3-cyano-3-methyl-7-methoxychroman-4-one and the structure of an “abnormal” product

A synthesis of 3-cyano-3-methyl-7-methoxychroman-4-one is reported. The structure of an “abnormal” product obtained during isomerization (III) with potassium t-butoxide in t-butanol, followed by alkylation with methyl iodide has been proved to be 3-t-butoxy-2-cyano- 2-mehthyl-2′,4′-dimethoxypropiophenone (IVa).

A revised structure for the condensation product of 2-carbethoxycyclopentanone and ethyl cyanoacetate

The condensation product of 2-carbethoxycyclopentanone and ethyl cyanoacetate is ethyl 2-carbethoxycyclopentylidene cyanoacetate (IIa) and not the one described by Kon and Nanji. Similarly, 2-carbomethoxycyclopentanone and methyl cyanoacetate yield methyl 2-carbomethoxycyclopentylidene cyanoacetate (IIb). The by-products obtained in the first reaction are cyclopentylidene cyanoacetate (IV) and the enamine of 2-carbethoxycyclopentanone (VIa).

Studies in sesquiterpenes—XXVI : Synthesis of 2,2,6-trimethyl-3-oxo-1-carbomethoxybicyclo[0,3,3]octane, a degradation product of patchouli alcohol

The synthesis of the title compound is described and results of some experiments on the degradation of patchouli alcohol are reported.

Synthetic investigations in the sesamin group - Part I. Synthesis of samin, a degradation product of sesamolin

Ethylα-bromovinylacetate (VII) was condensed with the sodio derivative of ethyl piperonoylacetate (VIII) to give diethylα-vinyl-α′-piperonoylsuccinate (IX). The latter on reduction with lithium aluminium hydride furnished the triol (X), which underwent smooth cyclisation with 1% ethanolic hydrogen chloride to 2-(3′, -methylenedioxyphenyl)-hydroxymethyl-4-vinyltetrahydrofuran (XIa). The structure of XIa was established by Oppenauer oxidation to an aldehyde. Ozonolysis of XIa afforded samin (I).

Synthesis of 1,3-dimethyl-1,3-dicarboxycyclohexane-2-acetic acid. An isomer of the degradation product of abietic acid

A synthesis of 1,3-dimethyl-1,3-dicarboxycyclohexane-2-acetic acid has been described, and proved to be an isomer of the C12-acid-an oxidative degradation product of abietic acid.

Reaction of dialkyl 2-butynoate with aniline and formaldehyde: revision of the structure of the product

Dimethyl 3-(aryl)-3,6-dihydro-2H-1,3-oxazine4,5-dicarboxylate structure assigned for the products obtained in the Bronsted acid catalyzed reaction of dimethyl but-2-ynoates with anilines and an excess of formaldehyde in methanol has been revised to methyl 1-(aryl)-3-(methoxymethyl)-4,5-dioxopyrrolidine-3-carboxylate. (C) 2010 Elsevier Ltd. All rights reserved.

An Eventful Synthetic Approach Towards the Biologically Potent Natural Product Ottelione A: Enantio-, Regio- and Stereoselective Construction of the Bicyclic Core

A novel synthetic approach towards the recently reported anti-tumor and anti-tuberculor natural product ottelione A from the readily available Diels-Alder adduct of cyclopentadiene and p-benzoquinone is delineated. Our short strategy, besides being enantio-, regio- and stereoselective, charts an eventful course and is inherently well-suited for adaptation towards diverse synthetic analogues of this biologically potent natural product.