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.