Let ℱn be the set of random mappings ϕ : {1,…,n} → {1,…,n} (such that every mapping is equally likely). For x ε {l,…,n} the elements are called the predecessors of x. Let Nr denote the random variable which counts the number of points x ε {l,…,n} with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n → ∞. If r = r(n) = o(n⅔) then the limiting distribution is Gaussian, if r ˜ Cn⅔ then it is Poisson, and in the remaining case rn−⅔ → ∞ it is degenerate. Furthermore, it is shown that Nr is a Poisson approximation if r → ∞.

We consider the problem of broadcasting in an n–node hypercube whose links and nodes fail independently with given probabilities p < 1 and q < 1, respectively. Information held in a fault-free node, called the source, has to reach all other fault-free nodes. Messages may be directly transmitted to adjacent nodes only, and every node may communicate with at most one neighbour in a unit of time. A message can be transmitted only if both communicating neighbours and the link joining them are fault-free. For parameters p and q satisfying (1 – p)(1 – q) ≽ 0.99 (e.g. p = q = 0.5%), we give an algorithm working in time O(log n) and broadcasting source information to all fault-free nodes with probability exceeding 1 – cn-e for some positive constant ε, c depending on p and q but not depending on n.

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

Let be the hypergraph whose points are the subsets X of [n] := {1,…,n} with l≤ |X| ≤ u, l < u, and whose edges are intervals in the Boolean lattice of the form I = {C ⊆[n] : X⊆C⊆Y} where |X| = l, |Y| = u, X ⊆ Y.We study the matching number i.e. the the maximum number of pairwise disjoint edges, and the covering number i.e. the minimum number of points which cover all edges. We prove that max and that for every ε > 0 the inequalities hold, where for the lower bounds we suppose that n is not too small. The corresponding fractional numbers can be determined exactly. Moreover, we show by construction that

Suppose each vertex of a bipartite multigraph (with partition (X, Y)) is assigned a set of colours; we say this colour scheme is feasible if the edges of the graph can be properly coloured so that each receives a colour in both of its endpoints' sets. We prove various results showing that certain types of colour scheme are always feasible. For instance, we prove that if the colour scheme obtained by assigning the set {1,…, d(x)} of colours to each vertex x of X and the set T = {1,…, t} (t < Δ(X)) to each vertex of Y is feasible, then so is every colour scheme where each vertex x of X gets d(x) colours from T and each vertex of Y gets the set T.

Let A be a subset of the lattice Ñ x N. We answer a question of Sárközy, proving if A is well distributed then the set of subset sums of A contains an infinite arithmetic progression.

In this paper, we present a techique for examining all trees of a given order. Our approach is based on the Beyer and Hedetniemi algorithm for generating all rooted trees of a given order and on the Wright, Richmond, Odlyzko and McKay algorithm for generating all free trees of a given order. In the introduction we describe these algorithms. We also give a precise evaluation of the average number of moves it takes to generate a rooted tree, which improves the upper bound given by Beyer and Hedetniemi. In the second section we present a new method of examining all trees which uses these generating algorithms. The last section contains two applications of the method introduced. The main result of the paper is that the average number of steps required by the proposed algorithm to examine a rooted tree is bounded by a constant independent of the order of a tree.

A connection is made between the theory of ergodicity and the expected complexity of string searching. In particular, a substring search algorithm is introduced which, when applied to searching in text that has been produced by an appropriate stationary ergodic source, has an expected running time of O((N/m + m)logm), for a text string of length N and search string of length m. Similar expected complexity results have been obtained before, but the analysis is performed in a significantly more general framework, which models with greater accuracy the statistics of many types of strings, including natural language. The analysis also sheds light on the performance of the Boyer-Moore algorithm and the Sunday algorithm when applied to natural language.

It is shown that, for every integer v < 7, there is a connected graph in which some v longest paths have empty intersection, but any v – 1 longest paths have a vertex in common. Moreover, connected graphs having seven or five minimal sets of longest paths (longest cycles) with empty intersection are presented. A 26-vertex 2-connected graph whose longest paths have empty intersection is exhibited.

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.