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.

We prove that there are at most {cA1/3} different lattice polygons of area A. This improves a result of V. I. Arnol'd.

The thickness of sparse random graphs in the model Gn, p is closely related to the arboricity, provided p(n) is suitably small. This allows us to identify a range of p(n) for which the thickness is approximately np/2.

In the Penney Ante game two players choose one binary string of length k each in turn, and toss a coin repeatedly. If at some stage the last k outcomes match one of their strings, the player with that string wins. The case k ≤ 4 is somewhat exceptional and in any case easily done. For k ≥ 5, Guibas and Odlyzko proved that the second player's optimal strategy is to choose the first k − 1 digits of the first player's string prefixed by 0 or 1. They conjectured that these two choices are never equally good. We prove that this conjecture is correct. Then we prove that 01…100 (with k − 1 ones) is an optimal strategy for the first player, and find all the strategies that are equally good.

Using a group-theoretic approach, we derive some Erdős-Ko-Rado-type results for certain Sperner families of chains and antichains in partial orders. In particular, we establish Bollobás-type inequalities for arbitrary Sperner families of intersecting affine subspaces, and special intersecting Sperner families in generalized Boolean algebras.

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

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.

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.