We consider a «Maker-Breaker’ version of the Ramsey Graph Game, RG(n), and present a winning strategy for Maker requiring at most (n − 3)2n−1 + n + 1 moves. This is the fastest winning strategy known so far. We also demonstrate how the ideas presented can be used to develop winning strategies for some related combinatorial games.

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.

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.

We show distributional results for the length of the longest matching consecutive subsequence between two independent sequences A1, A2, …, Am and B1, B2, …, Bn whose letters are taken from a finite alphabet. We assume that A1, A2, … are i.i.d. with distribution μ and B1, B2, … are i.i.d. with distribution ν. It is known that if μ and v are not too different, the Chen–Stein method for Poisson approximation can be used to establish distributional results. We extend these results beyond the region where the Chen–Stein method was previously successful. We use a combination of ‘matching by patterns’ results obtained by Arratia and Waterman [1], and the Chen–Stein method to show that the Poisson approximation can be extended. Our method explains how the matching is achieved. This provides an explanation for the formulas in Arratia and Waterman [1] and thus answers one of the questions posed in comment F19 in Aldous [2]. Furthermore, in the case where the alphabet consists of only two letters, the phase transition observed by Arratia and Waterman [1] for the strong law of large numbers extends to the distributional result. We conjecture that this phase transition on the distributional level holds for any finite alphabet.

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. We prove that any graph G of minimum degree at least three contains a dominating set D of size at most 3|V(G)|/8. A star S is a graph consisting of a centre x and a set of edges from x to S — x. Clearly, a dominating set D for a graph G corresponds to a set of |D| stars which cover V(G). Thus, we show that the vertices of any graph G of minimum degree 3 can be covered by at most 3|V(G)|/8 vertex disjoint stars. We also show that any connected cubic graph G can be covered by [|V(G)|/9] vertex disjoint paths. Both these results are sharp.

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.

Recently, Galvin [7] proved that every k-edge-colourable bipartite multigraph is k-edge-choosable. In particular, for a bipartite multigraph G, . Here we give a brief self-contained proof of this result.

We consider transition operators P on a countable set, which are reversible, irreducible and invariant under a group G of permutations of X with compact point stabilizers. We relate the computation of the spectral radius (norm) of P with the spectral radii of certain matrices defined over the factor set G\X. In various cases, this allows easy computation of the norm of P.

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.

In 1964 Dirac conjectured that every graph with n vertices and at least 3n − 5 edges contains a subdivision of K5 We prove a weakened version with 7/2;n − 7 instead of 3n − 5. We prove that, for any prescribed vertex νo, the subdivision can be found such that νo is not one of the five branch vertices. This variant of Dirac's problem, which was suggested by the present author in 1973, is best possible in the sense that 7/2;n − 7 cannot be replaced by 7/2;n − 15/2;.

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

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.

This paper deals with infinite binary sequences. Each sequence is treated as generated by a nondeterministic shift register. A measure-theoretic criterion helpful in finding a deterministic generator of the set of sequences is proposed.

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.