Let Per f(n) denote the set of all perfect graphs on n vertices and let Berge(n) denote the set of all Berge graphs on n vertices. The strong perfect graph conjecture states that Per f(n) = Berge(n) for all n. In this paper we prove that this conjecture is at least asymptotically true, i.e. we show that

We derive identities for the probability that at least a1 and at least a2, and for the probability that exactly a1 and exactly a2, out of n and N events occur (1 ≤ a1 ≤ n, 1 ≤ a2 ≤ N). From this, we produce multivariate permutation hybrid upper bounds, and a multivariate Bonferroni-type upper bound which includes Galambos and Xu's [2] optimal result. The methodology generalizes that of Hoppe and Seneta [3, §5]. A numerical example is given.

In this paper, we prove that every graph contains a cycle intersecting all maximum independent sets. Using this, we further prove that every graph with stability number α is spanned by α disjoint cycles. Here, the empty set, the graph of order 1 and the path of order 2 are all considered as degenerate cycles.

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

In this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

It is shown that every L2-summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L2-summand vectors of a dual real Banach space can be determined by the L2-summand vectors of its predual; for every n ∈ , every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.

Invariant submanifolds of contact (κ, μ)-manifolds are studied. Our main result is that any invariant submanifold of a non-Sasakian contact (κ, μ)-manifold is always totally geodesic and, conversely, every totally geodesic submanifold of a non-Sasakian contact (κ, μ)-manifold, μ ≠ 0, such that the characteristic vector field is tangent to the submanifold is invariant. Some consequences of these results are then discussed.

In this paper, some min–max theorems for even and C1 functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

We consider a linear integrodifferential equation of second order in a Hilbert space and show that the solution tends to zero polynomially if the decay of the convolution kernel is polynomial. Both polynomials are of the same order.

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry–Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

Let Mm be a closed smooth manifold with an involution having fixed point set of the form Fn ∪ F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n ≥ 4 is even (n < m). Suppose that the normal bundle of F2 in Mm, μ → F2, does not bound, and denote by β the stable cobordism class of μ → F2. In this paper, we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.

In this paper, we study ovals of symmetries and the fixed points of their products on Riemann surfaces of genus g ≥ 2. We show how the number of these points affects the total number of ovals of symmetries. We give a generalisation of Bujalance, Costa and Singerman's theorems in which we show upper bounds for the total number of ovals of two symmetries in terms of g, the order n and the number m of the fixed points of their product, and we show their attainments for n holding some divisibility conditions. Finally, we give an upper bound for m in terms of n and g, and we study conditions under which it has given parity.

For a function u harmonic in the unit disk , there holds the inequalitywhere p > 0 and β > −1, and.

We show that for each α>0 every sufficiently large oriented graph G with δ+(G), δ−(G)≥3|G|/8+α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G)+δ+(G)+δ−(G)≥3|G|/2 + α|G|. Up to the term α|G|, this confirms a conjecture of Häggkvist [10]. We also prove an Ore-type theorem for oriented graphs.