203 research outputs found

    Dirac's theorem for random regular graphs

    Get PDF
    We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever dd is sufficiently large compared to ε>0\varepsilon>0, a.a.s. the following holds: let GG' be any subgraph of the random nn-vertex dd-regular graph Gn,dG_{n,d} with minimum degree at least (1/2+ε)d(1/2+\varepsilon)d. Then GG' is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that dd 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

    Full text link
    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 (s,t)(s, t)-bipartite-hole in a graph GG consists of two disjoint sets of vertices SS and TT with S=s|S|= s and T=t|T|=t such that there are no edges between SS and TT; and α~(G)\widetilde{\alpha}(G) is the maximum integer rr such that GG contains an (s,t)(s, t)-bipartite-hole for every pair of non-negative integers ss and tt with s+t=rs + t = r. Our central theorem is that a graph GG with at least 33 vertices is Hamiltonian if its minimum degree is at least α~(G)\widetilde{\alpha}(G). 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 kk edge-disjoint Hamilton cycles. We see that for dense random graphs G(n,p)G(n,p), the probability of failing to contain many edge-disjoint Hamilton cycles is (1p)(1+o(1))n(1 - p)^{(1 + o(1))n}. Finally, we discuss the complexity of calculating and approximating α~(G)\widetilde{\alpha}(G)

    Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs

    Full text link
    P\'osa's theorem states that any graph GG whose degree sequence d1dnd_1 \le \ldots \le d_n satisfies dii+1d_i \ge i+1 for all i<n/2i < n/2 has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs GG of random graphs, i.e. we prove a `resilience version' of P\'osa's theorem: if pnClognpn \ge C \log n and the ii-th vertex degree (ordered increasingly) of GGn,pG \subseteq G_{n,p} is at least (i+o(n))p(i+o(n))p for all i<n/2i<n/2, then GG 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 Gn,pG_{n,p} 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

    Full text link
    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 nn vertices with minimum degree at least n/2n/2 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 pp, and prove that there exists a constant CC such that if pClogn/np \ge C \log n / n, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a (1:b)(1:b) Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias bb is at most cn/logncn /\log n for some absolute constant c>0c > 0. Both of these results are tight up to a constant factor, and are proved under one general framework.Comment: 36 page

    Bibliographie

    Get PDF

    Master index of volumes 161–170

    Get PDF

    Semi-Parametric Likelihood Functions for Bivariate Survival Data

    Get PDF
    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

    Get PDF
    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
    corecore