41,368 research outputs found
Existence of DĪ»-cycles and DĪ»-paths
A cycle of C of a graph G is called a DĪ»-cycle if every component of G ā V(C) has order less than Ī». A DĪ»-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 DĪ»-cycle or DĪ»-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
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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
Effective hamiltonian approach and the lattice fixed node approximation
We define a numerical scheme that allows to approximate a given Hamiltonian
by an effective one, by requiring several constraints determined by exact
properties of generic ''short range'' Hamiltonians. In this way the standard
lattice fixed node is also improved as far as the variational energy is
concerned. The effective Hamiltonian is defined in terms of a guiding function
and can be solved exactly by Quantum Monte Carlo methods. We argue
that, for reasonable and away from phase transitions, the long
distance, low energy properties are rather independent on the chosen guiding
function, thus allowing to remove the well known problem of standard
variational Monte Carlo schemes based only on total energy minimizations, and
therefore insensitive to long distance low energy properties.Comment: 8 pages, for the proceedings of "The Monte Carlo Method in the
Physical Sciences: Celebrating the 50th Anniversary of the Metropolis
Algorithm", Los Alamos, June 9-11, 200
Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model
A comprehensive study of the two-dimensional (2D) compass model on the square
lattice is performed for classical and quantum spin degrees of freedom using
Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art
implementations using Metropolis, stochastic series expansion and parallel
tempering techniques to obtain the critical ordering temperatures and critical
exponents. In a pre-investigation we reconsider the classical compass model
where we study and contrast the finite-size scaling behavior of ordinary
periodic boundary conditions against annealed boundary conditions. It is shown
that periodic boundary conditions suffer from extreme finite-size effects which
might be caused by closed loop excitations on the torus. These excitations also
appear to have severe effects on the Binder parameter. On this footing we
report on a systematic Monte Carlo study of the quantum compass model. Our
numerical results are at odds with recent literature on the subject which we
trace back to neglecting the strong finite-size effects on periodic lattices.
The critical temperatures are obtained as and
for the classical and quantum version, respectively,
and our data support a transition in the 2D Ising universality class for both
cases.Comment: 8 pages, 7 figures, differs slightly from published versio
Optimized local modes for lattice dynamical applications
We present a new scheme for the construction of highly localized lattice
Wannier functions. The approach is based on a heuristic criterion for
localization and takes the symmetry constraints into account from the start. We
compare the local modes thus obtained with those generated by other schemes and
find that they also provide a better description of the relevant vibrational
subspace.Comment: 6 pages, ReVTeX, plus four postscript files for figure
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