698 research outputs found
On powers of tight Hamilton cycles in randomly perturbed hypergraphs
We show that for , and , there exists
such that if and is a -uniform hypergraph
on vertices with minimum codegree at least , then asymptotically
almost surely the union contains the power of a
tight Hamilton cycle. The bound on is optimal up to the value of
and this answers a question of Bedenknecht, Han, Kohayakawa and
Mota
Random perturbation of sparse graphs
In the model of randomly perturbed graphs we consider the union of a deterministic graph Gα with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in Gα ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. Gα ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p)
A quantum Monte Carlo algorithm for Bose-Hubbard models on arbitrary graphs
We propose a quantum Monte Carlo algorithm capable of simulating the
Bose-Hubbard model on arbitrary graphs, obviating the need for devising
lattice-specific updates for different input graphs. We show that with our
method, which is based on the recently introduced Permutation Matrix
Representation Quantum Monte Carlo [Gupta, Albash and Hen, J. Stat. Mech.
(2020) 073105], the problem of adapting the simulation to a given geometry
amounts to generating a cycle basis for the graph on which the model is
defined, a procedure that can be carried out efficiently and and in an
automated manner. To showcase the versatility of our approach, we provide
simulation results for Bose-Hubbard models defined on two-dimensional lattices
as well as on a number of random graphs.Comment: 10 pages, 6 figure
Karp's patching algorithm on random perturbations of dense digraphs
We consider the following question. We are given a dense digraph with
minimum in- and out-degree at least , where is a constant.
We then add random edges to to create a digraph . Here an edge
is placed independently into with probability where
is a small positive constant. The edges of are given edge
costs , where is an independent copy of the exponential
mean one random variable i.e. . Let
be the associated cost matrix where
if . We show that w.h.p. the patching
algorithm of Karp finds a tour for the asymmetric traveling salesperson problem
that is asymptotically equal to that of the associated assignment problem.
Karp's algorithm runs in polynomial time.Comment: Fixed the proof of a lemm
Rainbow subgraphs of uniformly coloured randomly perturbed graphs
For a given , the randomly perturbed graph model is defined
as the union of any -vertex graph with minimum degree and
the binomial random graph on the same vertex set. Moreover,
we say that a graph is uniformly coloured with colours in if each
edge is coloured independently and uniformly at random with a colour from
.
Based on a coupling idea of McDiarmird, we provide a general tool to tackle
problems concerning finding a rainbow copy of a graph in a uniformly
coloured perturbed -vertex graph with colours in . For
example, our machinery easily allows to recover a result of Aigner-Horev and
Hefetz concerning rainbow Hamilton cycles, and to improve a result of
Aigner-Horev, Hefetz and Lahiri concerning rainbow bounded-degree spanning
trees.
Furthermore, using different methods, we prove that for any and integer , there exists such that the
following holds. Let be a tree on vertices with maximum degree at most
and be an -vertex graph with . Then a
uniformly coloured with colours in
contains a rainbow copy of with high probability. This is optimal both in
terms of colours and edge probability (up to a constant factor).Comment: 22 pages, 1 figur
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