Summary A geometric graph is a graph drawn in the plane such that its vertices are points in general position and its edges are straight-line segments. The study of geometric graphs is a fairly new discipline abounding in open problems, but it has already yielded some striking results that have proved to be instrumental for the solution of various problems in combinatorial and computational geometry. These include the k-set problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. This paper surveys some Turán-type and Ramsey-type extremal problems for geometric graphs, and discusses their generalizations and applications.

Introduction, basic definitions

Let G be a finite graph with no loops or multiple edges, whose vertex set and edge set are denoted by V(G) and E(G), respectively. By a drawing of G we mean a representation of G in the plane such that each vertex is represented by a distinct point and each edge by a simple (non-self-intersecting) continuous arc connecting the corresponding two points. If it is clear that we are referring to a drawing, and not to the underlying “abstract” graph, these points and arcs will also be called vertices and edges, respectively.

Two edges (arcs) cross each other if they have an interior point in common. This point is called a crossing. A crossing p is called proper if in a small neighbourhood of p one edge passes from one side of the other edge to the other side.

Summary Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we re-examine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined: in perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area: these are results which imply that there are no polynomial time algorithms of a particular type for a particular problem.

Introduction

The focus of this paper is approximate sampling and approximate counting (or approximate integration), using the Markov chain Monte Carlo (MCMC) method, and viewed from the perspective of combinatorial algorithms. There has been much work in this area in recent years, some of which we survey below in Section 4. We illustrate this work with a closer examination of one particular technique which has proved successful recently, that of coupling. This is a classical method from applied probability, but its application in this area has involved some new insights.

Formal foundations for work in this area were provided in the seminal paper of Jerrum, Valiant and Vazirani [50]. However, the subject seems subsequently to have outgrown the framework it provided. The present paper makes a modest attempt to update the situation.

Summary A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger's conjecture on t-colourability of graphs with no Kt+1 minor, Tutte's edge 3-colouring conjecture on edge 3-colourability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colourability of hypergraphs, and sign-nonsingular matrices.

Introduction

All graphs in this paper are finite, and may have loops and parallel edges. A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An H minor is a minor isomorphic to H. The following is Wagner's reformulation [75] of Kuratowski's theorem [27].

Theorem 1.1A graph is planar if and only if it has no minor isomorphic to K5 or K3,3.

Kuratowski's theorem is important, because it gives a good characterization (in the sense of J. Edmonds) of planarity, but we can also think of it as a structural theorem characterizing graphs with no K5 or K3,3 minor.

Summary This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are also presented, as well as various generalisations to random graphs with given degree sequence. A major feature in this area is the small subgraph conditioning method. When applicable, this establishes a relationship between random regular graphs with uniform distribution, and non-uniform models of random regular graphs in which the probability of a graph G is weighted according to the number of subgraphs that G has of a certain type. Information can be obtained in this way on the probability of existence of various types of spanning subgraphs, such as Hamilton cycles and decompositions into perfect matchings. Uniformly distributed labelled random regular graphs receive most of the attention, but also included are several non-uniform models which come about in a natural way. Some of these appear as spin-offs from the small subgraph conditioning method, and some arise from algorithms which use simple approaches to generating random regular graphs. A quite separate role played by algorithms is in the derivation of random graph properties by analysing the performance of an appropriate greedy algorithm on a random regular graph. Many open problems and conjeetures are given.

Summary We survey some parity arguments and problems in graph theory, in particular some that can be attacked using the cycle space of a graph. We discuss some results on specific collections of cycles that generate the cycle space. We explain how the space generated by the cycles through two prescribed edges in a graph is used in a proof of the conjecture made by B. Toft in 1974 that every 4-chromatic graph contains a totally odd K4-subdivision, that is, a subdivision of K4 in which each edge of K4 corresponds to an odd path. (Another proof of Toft's conjecture was found independently by W. Zang). We prove the new result that every 4-connected graph with at least three triangles contains a totally odd K4-subdivision if and only if it does not contain a vertex whose deletion results in a bipartite graph. In particular, every 4-connected planar graph contains a totally odd K4-subdivision. Finally, we offer some conjectures on path systems and subdivisions with parity constraints on the lengths.

Introduction

Parity arguments are often both elegant and powerful. An early parity result in graph theory is Redei's theorem [12] saying that the number of directed Hamiltonian paths in any tournament is odd. It implies, in particular, that every tournament has a directed Hamiltonian path. While this is an easy exercise, Redei's theorem inspired Forcade [4] to a parity result where the corresponding existence result is highly nontrivial.

Summary We study the following two problems in finite projective, affine and polar spaces of dimension d. Given integers 0 ≤ s, t ≤ d what is the cardinality of the smallest set T consisting of subspaces of dimension t with the property that every subspace of dimension s is incident with at least one element of T? Likewise, what is the cardinality of the largest set T consisting of subspaces of dimension t with the property that every subspace of dimension s is incident with at most one element of T? Bose and Burton solved the case t = 0 of the first question for projective spaces in 1966. We survey the known results, present some open problems, and prove new results for finite polar spaces.

Introduction

Consider a finite projective, affine or polar space of dimension d and two integers s and t with 0 ≤ s, t ≤ d and the following problems.

Problem 1: What is the smallest cardinality of a set T of t-subspaces such that every s-subspace is incident with at least one element of T?

Problem 2: What is the largest cardinality of a set T of t-subspaces such that every s-subspace is incident with at most one element of T?

Much attention has been paid to particular cases of both problems in the last 30 years. This article gives a survey and several new results concerning Problem 1. Some results concerning Problem 2 are mentioned too.

Summary A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs and unitals. In the case of nuclei these methods give lower bounds on the number of nuclei to a set of points in PG(n, q), usually dependent on some binomial coefficient not vanishing modulo the characteristic of the field. These lower bounds on nuclei lead directly to lower bounds on affine blocking sets with respect to lines. A short description of how linear polynomials can be used to construct maximal arcs in certain translation planes is included. A proof of the non-existence of maximal arcs in PG(2, q) when q is odd is outlined and some bounds are given as to when a (k, n)-arc can be extended to a maximal arc in PG(2, q). These methods can also be applied to unitals embedded in PG(2, q). One implication of this is that when q is the square of a prime a non-classical unital has a limited number of Baer sublines amongst its secants.

Introduction

The effectiveness of polynomials as a means of studying problems in finite geometries has become increasingly evident in the 1990's, although the first examples seem to date back to R. Jamison [38] in 1977 and A. E. Brouwer and A. A. Bruen and J. C. Fisher described the "Jamison method" as the following: reformulate the problem in terms of points of an affine space and associate suitable polynomials defined over the corresponding finite field; calculate. This is the approach employed in [19] too; in fact the main difference between [38] and [19] is that Jamison viewed the points of an affine space as elements of a finite field. In effect, this has the advantage of reducing the number of variables in the polynomials and allowing one to use simple arguments concerning the degree or the coefficients of a polynomial.

Summary Combinatorial designs have long had substantial application in the statistical design of experiments and in the theory of error-correcting codes. Applications in experimental and theoretical computer science have emerged, along with connections with the theory of cryptographic communication. This paper focuses on applications in the general area of communications, including cryptography and networking. Applications have been chosen to represent those in which design theory plays a useful, and sometimes central, role. Moreover, applications have been chosen to reflect in addition the genesis of new and interesting problems in design theory in order to treat the practical concerns. Of many candidates, thirteen applications areas have been included:

1. Optical orthogonal codes

2. Synchronous multiple access to channels

3. Group testing and superimposed codes

4. Erasure codes and information dispersal

5. Threshold and ramp schemes

6. Authentication codes

7. Resilient and correlation-immune functions

8. Multidrop networks

9. Channel graphs and interconnection networks

10. Partial match queries on files

11. Software testing

12. Disk layout and striping

13. (t, m, s)-nets and numerical integration

The theory of combinatorial designs continues to grow, in part as a consequence of the variety of these applications and the increasing depth of the connections with challenging problems on designs.

Summary The author rehearses his role in the development of the theory of matroids. The story starts in 1935 when he became an undergraduate at Trinity College, Cambridge, and started to collaborate with Leonard Brooks, Cedric Smith and Arthur Stone. It continues through his war-time work with codes and ciphers, followed by his return to Trinity in 1945, where his PhD thesis entitled “An Algebraic Theory of Graphs” foreshadowed his matroid papers published in 1958 and 1959. He describes the context in which he obtained the now well-known excluded minor conditions for a binary matroid to be regular and for a regular matroid to be graphic. He subsequently invented the whirl, and lectured on matroids at the 1964 Conference where the theory of matroids was first proclaimed to the world. This paper has two appendices: “Geometrical Terminology” and “Binary and Regular Matroids”.

As we all know, matroids made their appearance in the mathematical literature in 1935, in a paper of Hassler Whitney entitled “On the abstract properties of linear dependence” [17].

Whitney looked at a matrix and saw that some sets of columns were independent and some were not. There were even simple rules about this distinction. For example, any subset of an independent set of columns is independent—provided of course that you count the null set as independent. Also, if you had an independent set A you could make it into a bigger one by adding the right member of any independent set that was bigger than A.