An analytic study is made of the correlation structure of Tausworthe and linear congruential random number generators. The former case is analyzed by the bit mask correlations recently introduced by Compagner. The latter is studied first by an extension to word masks, which include spectral test coefficients as special cases, and then by the bit mask procedure. Although low order bit mask coefficients vanish in both cases, the Tausworthe generator appears to produce a substantially smaller non-vanishing correlation set for large masks – but with larger correlation values – than does the linear congruential.

There is a simple greedy algorithm for seeking large values of a function f defined on the vertices of the binary tree. Modeling f as a random function whose increments along edges are i.i.d., we show that (under a natural assumption) the values found by the greedy algorithm grow linearly in time, with rate specified in terms of a fixed-point identity for distributions.

The substitution method is used to show that the percolative behaviour of the triangular and hexagonal lattices bond percolation models are similar near their critical probabilities. As a consequence, if the limits defining the critical exponents β and γ exist, these lattices have the same values of β and γ. Similarly, the method also shows equality of the β and γ values for bond percolation models on the bowtie lattice and its dual.

An [n, k, r]-hypergraph is a hypergraph = (V, E) whose vertex set V is partitioned into n k-element sets V1, V2,…, Vn and for which, for each choice of r indices, 1 ≤ i1 < i2 < … < ir ≤ n, there is exactly one edge e ∈ E such that |e∩Vi| = 1 if i ∈ {i1, i2.…, ir} and otherwise |e ∩ Vi| = 0. An independent transversal of an [n, k, r]-hypergraph is a set T = {a1, a2,…, an} ⊆ V such that ai ∈ Vi for i = 1, 2, …, n and e ⊈ T for all e ∈ E. The purpose of this note is to estimate fr(k), defined as the largest n for which any [n, k, r]-hypergraph has an independent transversal. The sharpest results are for r = 2 and for the case when k is small compared to r.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

Consider an electrical network on n nodes with resistors rij between nodes i and j. Let Rij denote the effective resistance between the nodes. Then Foster's Theorem [5] asserts that

where i ∼ j denotes i and j are connected by a finite rij. In [10] this theorem is proved by making use of random walks. The classical connection between electrical networks and reversible random walks implies a corresponding statement for reversible Markov chains. In this paper we prove an elementary identity for ergodic Markov chains, and show that this yields Foster's theorem when the chain is time-reversible.

We also prove a generalization of a resistive inverse identity. This identity was known for resistive networks, but we prove a more general identity for ergodic Markov chains. We show that time-reversibility, once again, yields the known identity. Among other results, this identity also yields an alternative characterization of reversibility of Markov chains (see Remarks 1 and 2 below). This characterization, when interpreted in terms of electrical currents, implies the reciprocity theorem in single-source resistive networks, thus allowing us to establish the equivalence of reversibility in Markov chains and reciprocity in electrical networks.

Let and be sets of functions from domain X to ℝ. We say that validly generalises from approximate interpolation if and only if for each η > 0 and ∈, δ ∈ (0,1) there is m0(η, ∈, δ) such that for any function t ∈ and any probability distribution on X, if m > m0 then with m-probability at least 1 – δ, a sample X = (x1, X2,…,xm) ∈ Xm satisfies

We find conditions that are necessary and sufficient for to validly generalise from approximate interpolation, and we obtain bounds on the sample length m0{η,∈,δ) in terms of various parameters describing the expressive power of .

Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

A simple proof is given of the best-known upper bound on the cardinality of a set of vectors of length t over an alphabet of size b, with the property that, for every subset of k vectors, there is a coordinate in which they all differ. This question is motivated by the study of perfect hash functions.

Béla Bollobás [1] conjectured the following. For any positive integer Δ and real 0 < c < ½ there exists an n0 with the following properties. If n ≥ n0, T is a tree of order n and maximum degree Δ, and G is a graph of order n and maximum degree not exceeding cn, then there is a packing of T and G. Here we prove this conjecture. Auxiliary Theorem 2.1 is of independent interest.

This note contains a refinement of our paper [8], leading to an alternative proof of a conjecture of Mader and of Erdős and Hajnal recently proved by Bollobás and Thomason.

Suppose X1, X2,… is a sequence of independent and identically distributed random elements whose values are taken in a finite set S of size |S| ≥ 2 with probability distribution ℙ(X = s) = p(s) > 0 for s ∈ S. Pevzner has conjectured that for every probability distribution ℙ there exists an N > 0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1, X2,… has greater probability than that of A appearing before B. In this paper it is shown that a distribution ℙ satisfies Pevzner's conclusion if and only if the maximum value of ℙ, p, and the secondary maximum c satisfy the inequality . For |S| = 2 or |S| = 3, the inequality is true and the conjecture holds. If , then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mode.