We give short proofs of the following two results: Thomas's theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.

We show that recognizing the K3-freeness and K4-freeness of graphs is hard, respectively, for two-player nondeterministic communication protocols using exponentially many partitions and for nondeterministic syntactic read-r times branching programs.

The key ingredient is a generalization of a colouring lemma, due to Papadimitriou and Sipser, which says that for every balanced red—blue colouring of the edges of the complete n-vertex graph there is a set of εn2 triangles, none of which is monochromatic, such that no triangle can be formed by picking edges from different triangles. We extend this lemma to exponentially many colourings and to partial colourings.

We answer a question of Sós by showing that, if a graph G of order n and density p has no complete minor larger than would be found in a random graph G(n, p), then G is quasi-random, provided either p > 0.45631 … or κ(G) [ges ] n(log log log n)/(log log n), where 0.45631 … is an explicit constant.

The results proved can also be used to fill the gaps in an argument of Thomason, describing the extremal graphs having no Kt minor for given t.

The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

In this paper we study distances in random subgraphs of a generalized n-cube [Qscr ]ns over a finite alphabet S of size s. [Qscr ]ns is the direct product of complete graphs over s vertices, its vertices being the n-tuples (x1, …, xn), with xi ∈ S, i = 1, … n, and two vertices being adjacent if they differ in exactly one coordinate. A random (induced) subgraph γ of [Qscr ]ns is obtained by selecting [Qscr ]ns-vertices with independent probability pn and then inducing the corresponding edges from [Qscr ]ns. Our main result is that dγ (P,Q) [les ] [2k+3]d[Qscr ]ns (P,Q) almost surely for P,Q ∈ γ, pn = n−a and 0 [les ] a < ½, where k = [1+3a/1−2a] and dγ and d[Qscr ]ns denote the distances in γ and [Qscr ]ns, respectively.

A top to random shuffle of a deck of cards is performed by taking the top card off of the deck and replacing it in a randomly chosen position of the deck. We find approximations of the relative entropy of a deck of n cards after m successive top to random shuffles. Initially the relative entropy decays linearly and for larger m it decays geometrically at a rate that alters abruptly at m = n log n. It converges to an explicitly given expression when m = [n log n+cn] for a constant c.

The hexagonal lattice site percolation critical probability is shown to be at most 0.79472, improving the best previous mathematically rigorous upper bound. The bound is derived by using the substitution method to compare the site model with the bond model, the latter of which is exactly solved. Shortcuts which eliminate a substantial amount of computation make the derivation of the bound possible.