Hadwiger Number and the Cartesian Product Of Graphs

The Hadwiger number mr(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, mr(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 [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. 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 the (unique) prime factorization of G be given by G_1 [] G_2 [] ... [] G_k. 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 of Chandran and Sivadasan. 2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >= c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 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 (They had shown that the Hadiwger's conjecture is true for G^d if d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have been revised. The bounds are now asymptotically tigh

Boxicity of Series Parallel Graphs

The three well-known graph classes, planar graphs (P), series-parallel graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-parallel graph with boxicity 3, thus resolving this question. Recently Chandran and Sivadasan showed that for any G, box(G) <= treewidth(G)+2. They conjecture that for any k, there exists a k-tree with boxicity k+1. (This would show that their upper bound is tight but for an additive factor of 1, since the treewidth of any k-tree equals k.) The series-parallel graph we construct in this paper is a 2-tree with boxicity 3 and is thus a first step towards proving their conjecture.Comment: 10 pages, 0 figure

Synthesis of curcumin based imidazo[2,1-b]thiazole derivatives and their biological evaluation as antiproliferative agents

13-20Motivated by the antiproliferative potential of curcumin and imidazothiazoles, a series of curcumin based imidazo[2,1-b] thiazole derivatives have been prepared, characterized and evaluated for their anticancer activity against various human cancer cell lines. These synthesized compounds have been found to have appreciable to moderate activity. Consequently, compounds 8a and 8g display noteworthy cytotoxicity with IC50 values of 7.2 μM and 4.7 μM, respectively, against A549 cell line. Furthermore, compounds 8a, 8b and 8g exhibit substantial cytotoxicity with IC50 values ranging between 9.1 μM to 9.9 μM respectively, against HeLa cell line. Interestingly, compounds 8a and 8g exhibit appreciable cytotoxicity with IC50 values ranging between 7.5 μM to 8.7 μM respectively, against DU145 cancer cell line. Overall, four compunds (8a, 8b, 8g and 8h) demonstrate IC50 values less than 10 μM against selected human cancer cell lines. They could be taken further for investigation of their mode of action and other parameters

Synthesis and antimicrobial evaluation of benzothiazole linked isoxazole Schiff bases

1463-1470A new series of benzothiazole linked isoxazole Schiff base derivatives have been prepared and characterized by suitable spectroscopic methods via 1H and 13C NMR, ESI-MS and IR spectra. These compounds have been further screened for their antimicrobial activity against a panel of microorganisms. Among them, compounds 12d, 12g and 12l demonstrate promising antimicrobial activity against all the tested strains with MIC values ranging between 3.9 – 62.5 μg/mL. Further, compounds 12d, 12g and 12l exhibit promising antifungal activity with MIC values ranging between 7.8 – 32.5 μg/mL. Further studies are underway for determining the antifungal molecular mechanisms of these potential compounds

Synthesis and antimicrobial evaluation of benzothiazole linked isoxazole Schiff bases

A new series of benzothiazole linked isoxazole Schiff base derivatives have been prepared and characterized by suitable spectroscopic methods via 1H and 13C NMR, ESI-MS and IR spectra. These compounds have been further screened for their antimicrobial activity against a panel of microorganisms. Among them, compounds 12d, 12g and 12l demonstrate promising antimicrobial activity against all the tested strains with MIC values ranging between 3.9 – 62.5 µg/mL. Further, compounds 12d, 12g and 12l exhibit promising antifungal activity with MIC values ranging between 7.8 – 32.5 µg/mL. Further studies are underway for determining the antifungal molecular mechanisms of these potential compounds

Synthesis of curcumin based imidazo[2,1-b]thiazole derivatives and their biological evaluation as antiproliferative agents

Motivated by the antiproliferative potential of curcumin and imidazothiazoles, a series of curcumin based imidazo[2,1-b] thiazole derivatives have been prepared, characterized and evaluated for their anticancer activity against various human cancer cell lines. These synthesized compounds have been found to have appreciable to moderate activity. Consequently, compounds 8a and 8g display noteworthy cytotoxicity with IC50 values of 7.2 μM and 4.7 μM, respectively, against A549 cell line. Furthermore, compounds 8a, 8b and 8g exhibit substantial cytotoxicity with IC50 values ranging between 9.1 μM to 9.9 μM respectively, against HeLa cell line. Interestingly, compounds 8a and 8g exhibit appreciable cytotoxicity with IC50 values ranging between 7.5 μM to 8.7 μM respectively, against DU145 cancer cell line. Overall, four compunds (8a, 8b, 8g and 8h) demonstrate IC50 values less than 10 μM against selected human cancer cell lines. They could be taken further for investigation of their mode of action and other parameters

A concise and stereoselective synthesis of both enantiomers of altholactone and isoaltholactone

A concise and flexible stereoselective route to synthesize both enantiomers of the highly functionalized α,β-unsaturated-δ-lactones, altholactone and isoaltholactone, from readily available cinnamyl alcohol is described. This approach derived its asymmetry from Sharpless catalytic asymmetric epoxidation and Sharpless asymmetric dihydroxylation reactions. The resulting diols were produced in high enantiomeric excess and were cyclized in a stereoselective manner in the presence of a catalytic amount of camphor sulphonic acid. The synthesis of both enantiomers of altholactone and isoaltholactone has been achieved in a concise and highly enantioselective manner

Highly stereoselective synthesis of C-(alkynyl)-pseudoglycals from δ-hydroxy-α,β-unsaturated aldehydes

An efficient and novel methodology for the synthesis of C-(alkynyl)-pseudoglycals from δ-hydroxy-α,β-unsaturated aldehydes has been developed