438 research outputs found
Sparse Kneser graphs are Hamiltonian
For integers and , the Kneser graph is the
graph whose vertices are the -element subsets of and whose
edges connect pairs of subsets that are disjoint. The Kneser graphs of the form
are also known as the odd graphs. We settle an old problem due to
Meredith, Lloyd, and Biggs from the 1970s, proving that for every ,
the odd graph has a Hamilton cycle. This and a known conditional
result due to Johnson imply that all Kneser graphs of the form
with and have a Hamilton cycle. We also prove that
has at least distinct Hamilton cycles for .
Our proofs are based on a reduction of the Hamiltonicity problem in the odd
graph to the problem of finding a spanning tree in a suitably defined
hypergraph on Dyck words
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems
This is an introductory text reviewing Lieb-Robinson bounds for open and
closed quantum many-body systems. We introduce the Heisenberg picture for
time-dependent local Liouvillians and state a Lieb-Robinson bound that gives
rise to a maximum speed of propagation of correlations in many body systems of
locally interacting spins and fermions. Finally, we discuss a number of
important consequences concerning the simulation of time evolution and
properties of ground states and stationary states.Comment: 13 pages, 2 figures; book chapte
An entropy for groups of intermediate growth
One of the few accepted dynamical foundations of non-additive
"non-extensive") statistical mechanics is that the choice of the appropriate
entropy functional describing a system with many degrees of freedom should
reflect the rate of growth of its configuration or phase space volume. We
present an example of a group, as a metric space, that may be used as the phase
space of a system whose ergodic behavior is statistically described by the
recently proposed -entropy. This entropy is a one-parameter variation
of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from
the power-law entropies that have been recently studied. We use the first
Grigorchuk group for our purposes. We comment on the connections of the above
construction with the conjectured evolution of the underlying system in phase
space.Comment: 19 pages, No figures, LaTeX2e. Version 2: change of affiliation,
addition of acknowledgement. Accepted for publication to "Advances in
Mathematical Physics
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