It is well known that a matroid is a transversal matroid if and only if it is a matching matroid (in the sense that it is the restriction of the matching structure of some graph to a subset of its vertices). A simple proof of that result is now known and in this paper it is used to answer the long-standing question of which transversal matroids are “strict” matching matroids; i.e. actually equal to the matching structure of a graph. We develop a straightforward test of “coloop-surfeit” that can be applied to any transversal matroid, and our main theorem shows that a transversal matroid is a strict matching matroid if and only if it has even rank and coloop-surfeit. Furthermore, the proofs are algorithmic and enable the construction of an appropriate graph from any presentation of a strict matching matroid.

A deck of n cards is shuffled by repeatedly taking off the top m cards and inserting them in random positions. We give a closed form expression for the distribution after any number of steps. This is used to give the asymptotics of the approach to stationarity: for m fixed and n large, it takes shuffles to get close to random. The formulae lead to new subalgebras in the group algebra of the symmetric group.

If G is a plane, cubic graph, then G has a drawing such that each edge is a straight line segment and each bounded face has any prescribed area. The special case where all areas are the same proves a conjecture of G. Ringel, who gave an example of a plane triangulation that cannot be drawn in this way.

The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e., the time taken for it to reach its stationary or equilibrium distribution.

The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.

Due to recent developments in the area of computational formalisms for linguistic representation, the task of designing a parser for a specified natural language is now shifted to the problem of designing its grammar in certain formal ways. This paper describes the results of a project whose aim was to design a formal grammar for modern Hebrew. Such a formal grammar has never been developed before. Since most of the work on grammatical formalisms was done without regarding Hebrew (and other Semitic languages as well), we had to choose a formalism that would best fit the specific needs of the language. This part of the project has been described elsewhere. In this paper we describe the details of the grammar we developed. The grammar deals with simple, subordinate and coordinate sentences as well as interrogative sentences. Some structures were thoroughly dealt with, among which are noun phrases, verb phrases, adjectival phrases, relative clauses, object and adjunct clauses; many types of adjuncts; subcategorization of verbs; coordination; numerals, etc. For each phrase the parser produces a description of the structure tree of the phrase as well as a representation of the syntactic relations in it. Many examples of Hebrew phrases are demonstrated, together with the structure the parser assigns them. In cases where more than one parse is produced, the reasons of the ambiguity are discussed.

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

We discuss a hierarchical probabilistic model whose predictions are similar to those of the popular language modelling procedure known as ‘smoothing’. A number of interesting differences from smoothing emerge. The insights gained from a probabilistic view of this problem point towards new directions for language modelling. The ideas of this paper are also applicable to other problems such as the modelling of triphomes in speech, and DNA and protein sequences in molecular biology. The new algorithm is compared with smoothing on a two million word corpus. The methods prove to be about equally accurate, with the hierarchical model using fewer computational resources.

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.