Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

A Dirac type result on Hamilton cycles in oriented graphs

We show that for each \alpha>0 every sufficiently large oriented graph G with \delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if \delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term \alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type theorem for oriented graphs.Comment: Added an Ore-type resul

Geodesics on the Ellipsoid and Monodromy

The equations for geodesic flow on the ellipsoid are well known, and were first solved by Jacobi in 1838 by separating the variables of the Hamilton-Jacobi equation. In 1979 Moser investigated the case of the general ellipsoid with distinct semi-axes and described a set of integrals which weren't know classically. After reviewing the properties of geodesic flow on the three dimensional ellipsoid with distinct semi-axes, we investigate the three dimensional ellipsoid with the two middle semi-axes being equal, corresponding to a Hamiltonian invariant under rotations. The system is Liouville-integrable and thus the invariant manifolds corresponding to regular points of the energy momentum map are 3-dimensional tori. An analysis of the critical points of the energy momentum maps gives the bifurcation diagram. We find the fibres of the critical values of the energy momentum map, and carry out an analysis of the action variables. We show that the obstruction to the existence of single valued globally smooth action variables is monodromy.Comment: 24 pages, 7 figure

Spin dynamics in the Kapitza-Dirac effect

Electron spin dynamics in Kapitza-Dirac scattering from a standing laser wave of high frequency and high intensity is studied. We develop a fully relativistic quantum theory of the electron motion based on the time-dependent Dirac equation. Distinct spin dynamics, with Rabi oscillations and complete spin-flip transitions, is demonstrated for Kapitza-Dirac scattering involving three photons in a parameter regime accessible to future high-power X-ray laser sources. The Rabi frequency and, thus, the diffraction pattern is shown to depend crucially on the spin degree of freedom