Introduction

Graph coloring is arguably the most popular subject in graph theory. An interesting variant of the classical problem of coloring properly the vertices of a graph with the minimum possible number of colors arises when one imposes some restrictions on the colors available for every vertex. This variant received a considerable amount of attention that led to several fascinating conjectures and results, and its study combines interesting combinatorial techniques with powerful algebraic and probabilistic ideas. The subject, initiated independently by Vizing [51] and by Erdös, Rubin and Taylor [24], is usually known as the study of the choosability properties of a graph. In the present paper we survey some of the known recent and less recent results in this topic, focusing on the techniques involved and mentioning some of the related intriguing open problems. This is mostly a survey article, but it contains various new results as well.

Introduction

I will review certain results and problems concerning the covering of the edge set of a graph with circuits or Euler tours. The areas I shall consider are:

• Faithful circuit covers of weighted graphs.

• Shortest circuit covers of bridgeless graphs.

• Compatible circuit decompositions and Euler tours of Eulerian graphs.

• Even circuit decompositions of Eulerian graphs.

I will describe interconnections between these areas and also indicate connections with other areas of combinatorics: matroids, isotropic systems, edge colouring of graphs, latin squares, nowhere-zero flows in graphs, and the Chinese postman problem.

Throughout this survey my graphs will be finite and without loops, although they may contain multiple edges. By a closed walk in a graph G, I will mean an alternating sequence of vertices and edges which starts and ends at the same vertex and is such that consecutive vertices and edges are incident. The length of the walk is the number of edges in the sequence. A tour is a closed walk which does not repeat edges. An Euler tour of G is a tour which traverses every edge of G. We shall say that a graph is Eulerian if it has an Euler tour. A tour decomposition of G is a partition of E(G) into (edge sets of) tours. We will consider an Euler tour of G to be a tour decomposition into exactly one tour. A circuit is a tour which does not repeat vertices, except when the first vertex is repeated as the last.

Introduction.

The subject of the survey can be viewed in a general framework of local characterization of graphs. That is characterization by structure of the neighbourhoods of vertices. Usually this structure is described by the subgraph induced by the neighbourhood. If the vertex stabilizer in the automorphism group of the considered graph induces on the neighbourhood a doubly transitive permutation group the induced subgraph is trivial (either the complete graph or the null graph).

Introduction

Recent years have seen numerous interesting applications of combinatorics to cryptography. In particular, combinatorial designs have played an important role in the study of such topics in cryptography as secrecy and authentication codes, secret sharing schemes, and resilient functions. The purpose of this paper is to elucidate some of these connections. This is not intended to be an exhaustive survey, but rather a sampling of some research topics in which I have a personal interest.

Some other related surveys include applications of error-correcting codes to cryptography (Sloane [60]), applications of finite geometry to crpytography (Beutelspacher [6]) and applications of combinatorial designs to computer science (Colbourn and Van Oorschot [16]).

In this introduction we will define the relevant concepts from design theory that we will have occasion to use. Beth, Jungnickel and Lenz [4] is a good general reference on design theory.

Let 1 ≤ t ≤ k < v. An Sλ(t, k, v) is defined to be a pair (X, β), where X is a v-set (of points) and β is a collection of κ-subsets of X (called blocks), such that every t-subset of points occurs in exactly λ blocks. An Sλ(t, k, v) is called simple if no block occurs more than once in β.

INTRODUCTION

In 1960 C.St.J.A. Nash-Williams generalized the following easy but pretty result of H.E. Robbins [1939]: the edges of an undirected graph G can be oriented so that the resulting directed graph D is strongly connected if and only if G is 2-edge-connected.

To formulate the generalization let us call a directed graph [undirected graph] κ-edge-connected if there are κ edge-disjoint directed (undirected)) paths from each node to each other.)

WEAK ORIENTATION THEOREM 1.1 [Nash-Williams, 1960] The edges of an undirected graph G can be oriented so that the resulting directed graph is κ-edge-connected if and only if G is 2κ-edge-connected.

The neccessity of the condition is straightforward and the main difficulty lies in proving its sufficiency. Actually, Nash-Williams proved a stronger result. To formulate it, we need the following notation. Given a directed or undirected graph G, let λ(x, y; G) denote local edge-connectivity from x to y, that is, the maximum number of edge-disjoint paths from x to y.

Abstract. We introduce the concept of a weighted quasigroup. We show that, corresponding to any weighted quasigroup, there is a quasigroup from which it can be obtained in a certain natural way, which we term amalgamation. Not all commutative weighted quasigroups can be obtained from commutative quasigroups by amalgamation; however, given a commutative weighted quasigroup whose weights are all even, we give a necessary and sufficient condition for the existence of a commutative quasigroup from which it can be obtained by amalgamation. We also discuss conjugates of weighted quasigroups.

We also introduce the concept of a simplex zeroid, and relate this concept to that of a weighted quasigroup.

Weighted quasigroups.

Suppose that we have a finite set S with a closed binary operation. If a, b ε S, we shall denote the result of this binary operation acting on a and b by ab. If the binary operation has the two properties

1. (i) for each a, b ε S, the equation ax = b is uniquely solvable for x, and

2. (ii) for each a, b ε S the equation ya = b is uniquely solvable for y,

then S is a quasigroup. It is well-known, and easy to see, that S is a quasigroup if and only if its multiplication table is a latin square. The properties (i) and (ii) amount to the assertion that, in the multiplication table, each element of S occurs exactly once in each row and exactly once in each column.

Introduction

In this paper I will present a number of problems and for the most part recent results in combinatorics in general and finite geometry in particular. The property these problems share is the fact that they can all be attacked using some kind of polynomial trick. Unfortunately I am not able to give a characterization of the kind of problem that can be solved with these methods, but I am convinced that seeing how the polynomials work in a number of cases should give an impression of the type of problems that might be attacked.

The starting point is usually a combinatorial problem of the following form: Given a set of points (or vectors, or sets) that satisfy some property, we want to say something about the size or the structure of this set. The approach is then to associate to this set a polynomial, or a collection of polynomials, and use properties of polynomials to obtain information on the size or structure of the set.

The setup of this paper is roughly as follows. We consider a particular property of polynomials and give examples where this property can be used to attack the problem.

The fourteenth British Combinatorial Conference is to be held here in July 1993 and, as is usual, the British Combinatorial Committee has invited nine distinguished combinatorial mathematicians to give survey talks on the latest developments in their own fields. This volume contains the papers which they have agreed to submit in advance of their talks.

The quality of these contributions and the interest shown are encouraging signs that the fourteenth conference will be as successful as its predecessors.

The sixtieth birthday of Crispin Nash-Williams falls in 1993, and to mark his tremendous contribution to the growth of combinatorics both in Britain and the world, one day of the conference is to be ‘Nash-Williams day’. The talks by András Frank and Anthony Hilton on that day will be in his honour, as will be many of the contributed talks, and to him this volume is dedicated.

A special edition of Discrete Mathematics, edited by Douglas Woodall, will contain papers contributed to the conference.

I wish to thank the contributors and the referrees for their co-operation in meeting quite tight deadlines, Roger Astley of Cambridge University Press, and the London Mathematical Society and the Institute of Combinatorics and its Applications for their financial support.

Introduction

The theory of random graphs has proved to be a success. It has provided many interesting questions, provoking the development of new techniques on the boundary between combinatorics and probability theory. Random graphs have also proved tremendously valuable in applications outside the theory: firstly in extremal questions, where bounds obtained by nonconstructive means are often the best known, and secondly in questions coming from computer science, or, for instance, from biology, which turn out to translate directly into natural questions about particular models of random graphs. It is not appropriate to give an extensive bibliography here, but the place to set off on a serious study of random graphs is the monograph by Bollobas [7].

Another basic combinatorial structure is that of finite partially ordered sets, and it is very natural to ask whether an equally successful theory of random partial orders can be developed. Regrettably, the simple answer will have to be “no”. One reason for this is perhaps that partial orders occur less often in applications than do graphs, but the more important problem is mathematical: the lack of “independence”.

To explain this, it is convenient to recall first the most basic model of random graphs. Let n be a positive integer, and p a real number strictly between 0 and 1. Set [n] = { 1, …, n}. For each pair (i, j) of integers from [n] with i < j, let Xij be a Bernoulli random variable with Pv(Xij = 1) = p, with all the Xij mutually independent.