203 research outputs found
Dirac's theorem for random regular graphs
We prove a `resilience' version of Dirac's theorem in the setting of random
regular graphs. More precisely, we show that, whenever is sufficiently
large compared to , a.a.s. the following holds: let be any
subgraph of the random -vertex -regular graph with minimum
degree at least . Then is Hamiltonian.
This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result
is best possible: firstly, the condition that is large cannot be omitted,
and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability &
Computin
Hamilton cycles, minimum degree and bipartite holes
We present a tight extremal threshold for the existence of Hamilton cycles in
graphs with large minimum degree and without a large ``bipartite hole`` (two
disjoint sets of vertices with no edges between them). This result extends
Dirac's classical theorem, and is related to a theorem of Chv\'atal and
Erd\H{o}s.
In detail, an -bipartite-hole in a graph consists of two disjoint
sets of vertices and with and such that there are no
edges between and ; and is the maximum integer
such that contains an -bipartite-hole for every pair of
non-negative integers and with . Our central theorem is that
a graph with at least vertices is Hamiltonian if its minimum degree is
at least .
From the proof we obtain a polynomial time algorithm that either finds a
Hamilton cycle or a large bipartite hole. The theorem also yields a condition
for the existence of edge-disjoint Hamilton cycles. We see that for dense
random graphs , the probability of failing to contain many
edge-disjoint Hamilton cycles is . Finally, we discuss
the complexity of calculating and approximating
Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs
P\'osa's theorem states that any graph whose degree sequence satisfies for all has a Hamilton cycle.
This degree condition is best possible. We show that a similar result holds for
suitable subgraphs of random graphs, i.e. we prove a `resilience version'
of P\'osa's theorem: if and the -th vertex degree (ordered
increasingly) of is at least for all ,
then has a Hamilton cycle. This is essentially best possible and
strengthens a resilience version of Dirac's theorem obtained by Lee and
Sudakov.
Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree
sequences which ensure the existence of a Hamilton cycle. We show that a
natural guess for a resilience version of Chv\'atal's theorem fails to be true.
We formulate a conjecture which would repair this guess, and show that the
corresponding degree conditions ensure the existence of a perfect matching in
any subgraph of which satisfies these conditions. This provides an
asymptotic characterisation of all degree sequences which resiliently guarantee
the existence of a perfect matching.Comment: To appear in the Electronic Journal of Combinatorics. This version
corrects a couple of typo
Robust Hamiltonicity of Dirac graphs
A graph is Hamiltonian if it contains a cycle which passes through every
vertex of the graph exactly once. A classical theorem of Dirac from 1952
asserts that every graph on vertices with minimum degree at least is
Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend
Dirac's theorem in two directions and show that Dirac graphs are robustly
Hamiltonian in a very strong sense. First, we consider a random subgraph of a
Dirac graph obtained by taking each edge independently with probability ,
and prove that there exists a constant such that if ,
then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we
prove that if a Maker-Breaker game is played on a Dirac graph, then
Maker can construct a Hamiltonian subgraph as long as the bias is at most
for some absolute constant . Both of these results are
tight up to a constant factor, and are proved under one general framework.Comment: 36 page
Semi-Parametric Likelihood Functions for Bivariate Survival Data
Because of the numerous applications, characterization of multivariate survival distributions is still a growing area of research. The aim of this thesis is to investigate a joint probability distribution that can be derived for modeling nonnegative related random variables. We restrict the marginals to a specified lifetime distribution, while proposing a linear relationship between them with an unknown (error) random variable that we completely characterize. The distributions are all of positive supports, but one class has a positive probability of simultaneous occurrence. In that sense, we capture the absolutely continuous case, and the Marshall-Olkin type with a positive probability of simultaneous event on a set of measure zero. In particular, the form of the joint distribution when the marginals are of gamma distributions are provided, combining in a simple parametric form the dependence between the two random variables and a nonparametric likelihood function for the unknown random variable. Associated properties are studied and investigated and applications with simulated and real data are given
Collective excitations of the QED vacuum
Using relativistic Green’s-function techniques we examined single-electron excitations from the occupied Dirac sea in the presence of strong external fields. The energies of these excited states are determined taking into account the electron-electron interaction. We also evaluate relativistic transition strengths incorporating retardation, which represents a direct measure of correlation effects. The shifts in excitation energies are computed to be lower than 0.5%, while the correlated transition strengths never deviate by more than 10% from their bare values. A major conclusion is that we found no evidence for collectivity in the electron-positron field around heavy and superheavy nuclei
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