Hamilton cycles in highly connected and expanding graphs

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory.Comment: 19 page

A new class of two-layer Green-Naghdi systems with improved frequency dispersion

We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1+1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness and stability results. These results apply in particular to the original Green-Naghdi model as well as to the Saint-Venant (hydrostatic shallow-water) system with surface tension.Comment: to appear in Stud. Appl. Mat

Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Maximal Stability Regions for Superconducting Ground States of Generalized Hubbard Models

For a class of generalized Hubbard models, we determine the maximal stability region for the superconducting eta-pairing ground state. We exploit the Optimized Ground State (OGS) approach and the Lanczos diagonalization procedure to derive a sequence of improved bounds. We show that some pieces of the stability boundary are asymptotic, namely independent on the OGS cluster size. In this way, necessary and sufficient conditions are obtained to realize superconductivity in terms of an eta-pairing ground state. The phenomenon is explained by studying the properties of certain exact eigenstates of the OGS hamiltonians.Comment: 9 pages, 5 PostScript figures, submitted to Phys. Rev. Let