Given a monotone graphical property Q, how large should d(n) be to ensure that if (Hn) is any sequence of graphs satisfying |Hn| = n and (Hn)¿d(n), then almost every induced subgraph of Hn has property Q? We prove essentially best possible results for the following monotone properties: (i) k-connected for :xed k, (ii) Hamiltonian. c © 2002 Elsevier Science B.V. All rights reserved.

Let k ≥ 2 be a fixed integer and P be a property of k-uniform hypergraphs. In other words, P is a (typically infinite) family of k-uniform hypergraphs and we say a given hypergraph H satisfies P if H ∈ P . For a given constant η > 0 a k-uniform hypergraph H on n vertices is η-far from P if no hypergraph obtained from H by changing (adding or deleting) at most ηn edges in H satisfies P . More pr...

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum of two functions with the single crossing property need not have the same property. We obtain the precise condition...

Let C be an n-dimensional integral box, and π be a monotone property defined over the elements of C. We consider the problems of incrementally generating jointly the families Fπ and Gπ of all minimal subsets satisfying property π and all maximal subsets not satisfying property π, when π is given by a polynomial-time satisfiability oracle. Problems of this type arise in many practical applicatio...

Being motivated in terms of mathematical concepts from the theory of electrical networks, Klein & Ivanciuc introduced and studied a new graph-theoretic cyclicity index–the global cyclicity index (Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem. 30 (2001) 271–287). In this paper, by utilizing techniques from graph theory, electrical network theory and real analysis, we...

Let P be a monotone property of directed graphs on n vertices, and let P n denote the abstract simplicial complex whose simplices are the edge sets of graphs having property P. We prove that: 1. If \P = acyclic" then P n is homotopy equivalent to the (n ? 2)-sphere. 2. If \P = not strongly connected" then P n has the homotopy type of a wedge of (n ? 1)! spheres of dimension 2n ? 4. The lattice ...

Moody, P. J. and A. W. Roscoe, Acyclic monotone normality, Topology and its Applications 47 (1992) 53-67. A space X is acyclic monotonically normal if it has a monotonically normal operator M(., .) such that for distinct points x,,, ,x._, in X and x, =x,], n::i M(x,, X\{x,+,}) = (d. It is a property which arises from the study of monotone normality and the condition “chain (F)“. In this paper i...

In the investigation of boundary value problems the construction of a twosided inclusion of the solution can be as important as a numerical approximation of the solution itself. In the present paper we analyze a monotone discretization technique of higher order based upon piecewise interpolation and shifting such that bounding upper and lower solutions are obtained. The monotone discretization ...