This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr][Pscr]-complete, although it is in [Pscr] if κ is fixed.)

We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

In this paper, we study a statistical property of classes of real-valued functions that we call approximation from interpolated examples. We derive a characterization of function classes that have this property, in terms of their ‘fat-shattering function’, a notion that has proved useful in computational learning theory. The property is central to a problem of learning real-valued functions from random examples in which we require satisfactory performance from every algorithm that returns a function which approximately interpolates the training examples.

Answering a question of Wilf, we show that, if n is sufficiently large, then one cannot cover an n × p(n) rectangle using each of the p(n) distinct Ferrers shapes of size n exactly once. Moreover, the maximum number of pairwise distinct, non-overlapping Ferrers shapes that can be packed in such a rectangle is only Θ(p(n)/log n):

A cladogram is a tree with labelled leaves and unlabelled degree-3 branchpoints. A certain Markov chain on the set of n-leaf cladograms consists of removing a random leaf (and its incident edge) and re-attaching it to a random edge. We show that the mixing time (time to approach the uniform stationary distribution) for this chain is at least O(n2) and at most O(n3).

For any fixed l < k we present families of asymptotically good packings and coverings of the l-subsets of an n-element set by k-subsets, and an algorithm that, given a natural number i, finds the ith k-subset of the family in time and space polynomial in log n.

The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.

For a finite multigraph G, the reliability function of G is the probability RG(q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that, for any connected multigraph G, if q ∈ [Copf] is such that RG(q) = 0 then [mid ]q[mid ] [les ] 1. We verify that this conjectured property of RG(q) holds if G is a series-parallel network. The proof is by an application of the Hermite–Biehler theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown–Colbourn conjecture, and are hence true for cographic matroids of series-parallel networks.

For a subgroup W of the hyperoctahedral group On which is generated by reflections, we consider the linear dependence matroid MW on the column vectors corresponding to the reflections in W. We determine all possible automorphism groups of MW and determine when W ≅ = Aut(MW). This allows us to connect combinatorial and geometric symmetry. Applications to zonotopes are also considered. Signed graphs are used as a tool for constructing the automorphisms.

The core of a graph G is the subgraph GΔ induced by the vertices of maximum degree. We define the deleted core D(G) of G. We extend an earlier sufficient condition due to Hoffman [7] for a graph H to be the core of a Class 2 graph, and thereby provide a stronger sufficient condition. The new sufficient condition is in terms of D(H). We show that in some circumstances our condition is necessary; but it is not necessary in general.

Let Gp be a random graph on 2d vertices where edges are selected independently with a fixed probability p > ¼, and let H be the d-dimensional hypercube Qd. We answer a question of Bollobás by showing that, as d → ∞, Gp almost surely has a spanning subgraph isomorphic to H. In fact we prove a stronger result which implies that the number of d-cubes in G ∈ [Gscr](n, M) is asymptotically normally distributed for M in a certain range. The result proved can be applied to many other graphs, also improving previous results for the lattice, that is, the 2-dimensional square grid. The proof uses the second moment method – writing X for the number of subgraphs of G isomorphic to H, where G is a suitable random graph, we expand the variance of X as a sum over all subgraphs of H itself. As the subgraphs of H may be quite complicated, most of the work is in estimating the various terms of this sum.

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh(E maxi∥ai[bull]∥h + E maxj∥aj[bull]∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h(E maxi∥ai[bull]∥h + E maxj∥aj[bull]∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).