A simple and elementary proof of Whitney's unique embedding theorem

In this note we give a short and elementary proof of a more general version of Whitney's theorem that 3-connected planar graphs have a unique embedding in the plane. A consequence of the theorem is that cubic plane graphs cannot be embedded in a higher genus with a simple dual. The aim of this paper is to promote a simple and elementary proof, which is especially well suited for lectures presenting Whitney's theorem

The Generation of Fullerenes

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes -- fullgen -- and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.Comment: 21 pages; added a not

Heavier Group 2 Metals: Application to Intermolecular Hydroamination, C-F Activation and Intramolecular Hydroalkoxylation

This thesis describes the reactivity of different heavier alkaline earth catalysts [M{X(SiMe3)2}2(THF)n]m (M = Ca, Sr, Ba; X = N, CH; n= 0, 2; m= 1, 2) in the intermolecular hydroamination of styrene derivatives. The scope of these reactions with respect to the substrate was determined and detailed kinetic studies to establish rate law and temperature dependence of the hydroamination reactions reported were conducted. Overall, it was found that [Ca{N(SiMe3)2}2]2 is favoured enthalpically (Ca: ΔH‡ = 51 kJ∙mol-1, Sr: ΔH‡ = 71 kJ∙mol-1) however the corresponding strontium bis(amide) proved a significantly better catalyst, likely due to a favourably high entropy of activation value (Ca: ΔS‡ = -168 J/mol-1 ·K-1, Sr: ΔS‡ = -92 J∙mol-1∙K-1). Large kinetic isotope effects of 4.1 and 7.9 at 55 °C for the intermolecular hydroamination of styrene with piperidine mediated by [Ca{N(SiMe3)2}2]2 and [Sr{N(SiMe3)2}2]2, respectively, suggest a rate-determining alkene insertion into the M-N bond with immediate or concerted protonolysis. The methodology used in these hydroamination reactions was extended to simple dienes, diphenylacetylene and an activated enyne. The catalyst initiation of the metal bis(amides) with piperidine was shown to be reversible and the equilibrium constant solvent dependent. Novel calcium and strontium dialkyl complexes [M{CH(SiMe3)2}2(THF)2] (M= Ca, Sr) were used to overcome the problem of catalyst initiation and showed a different solvent dependence. An enhanced reactivity was found for the dialkyl complexes compared to the metal bis(amides). This increased reactivity allowed the application in new reactions such as the C-F activation of fluorobenzenes. Furthermore, the use of these catalytic systems was successfully extended to intramolecular hydroalkoxylation reactions of alkynyl alcohols in the formation of five- and six-membered enol ethers. In this case, [Ba{N(SiMe3)2}2]2 displayed significant reactivity although the “catalyst of choice” for these reactions proved to be strongly dependent on substrate substitution pattern. Through detailed kinetic studies the catalyst, substrate and temperature dependence of the cyclisation reaction were established and an unusual rate law with inverse substrate dependence proposed