Abstract

A double-five of planes is a set ψ of 35 points in PG(5, 2) which admits two distinct decompositions ψ = α1 ∪ α2 ∪ α3 ∪ α4 ∪ α5 = β1 ∪ β2 ∪ β3 ∪ β4 ∪ β5 into a set of five mutually skew planes such that αr ∩ βr is a line, for each r, while αr ∩ βs is a point, for r ≠ s. In a recent paper, [Sh96], a construction of a double-five was given, starting out from a (suitably coloured) icosahedron, and some of its main properties were described. The present paper deals first of all with some further properties of double-fives. In particular the existence of an invariant symplectic form is demonstrated and some related duality properties are described.

Secondly the relationship of double-fives to partial spreads of planes in PG(5, 2) is considered. The α-planes, or equally the β-planes, of double-fives provide the only examples of maximal partial spreads. It is shown that one of the planes of a non-maximal partial spread of five planes is always privileged, and this fact is seen to give rise to a nice geometric construction of an overlarge set of nine 3-(8, 4, 1) designs having automorphism group ΓL2(8).

Abstract

This is a survey of all known rank three geometries belonging to a string diagram of type (c*, c)-geometry. There are three types of objects: points, lines, and blocks subject to axioms imposed by the diagram. There are several other formulations described here which are more convenient for presenting certain of the examples. All examples fall into these six classes:

1. Simplicial type, which can easily be characterized.

2. Fischer spaces with no affine planes.

3. Orthogonal types, whose points and lines are exterior points and tangent lines of certain low-dimensional quadrics.

4. Hall type, determined by alternating multilinear forms over the field of two elements.

5. Affine type, whose points are vectors in some d-dimensional space over the integers mod 2. Here, blocks are not subspaces.

6. A few special examples determined by coherent pairs: the construction of odd type of Cameron and Fisher, and two examples of Blokhuis and Brouwer.

Abstract

New algorithms are presented for finding spreads and packings of sets with applications to combinatorial designs and finite geometries. An efficient deterministic method for spread enumeration is used to settle several existence problems for t-designs and partial geometries. Randomized algorithms based on tabu search are employed to construct new Steiner 5-designs and large sets of combinatorial designs. In particular, partitions are found of the 4-subsets of a 16-set into 91 disjoint affine planes of order 4.

Abstract

Buekenhout has given a construction of unitals in PG(2, q2) using the André representation of PG(2, q2) in the space PG(4,q). Metz has shown that this construction produces hermitian and non-hermitian unitals. In this note, we give a geometric criterion in PG(4, q) to decide whether the unital in PG(2, q2) is hermitian or not.

The 1997 issue of the British Combinatorial Bulletin contains a short history, written by Norman Biggs, of the early years of the British Combinatorial Conference. The first one was held at Oxford in 1969. The sixth conference, held at Royal Holloway College in 1977, was the first at which a volume containing the invited talks was published in time to be available to participants at the conference. Peter Cameron was the pioneering editor of that volume. Such a volume has been produced for every conference thereafter.

The 1977 conference was also the first one that I attended. There I joined the British Combinatorial Committee, which was formally set up at that meeting although it had effectively existed for some years—the previous conferences didn't just organize themselves. As often happens, I found that being on the committee considerably widened my knowledge of the subject. I left the committee in 1981, but have never lost touch with combinatorial activity in Britain.

I was delighted when I was asked to edit the present volume. In spite of the work involved, I am still delighted. I have had a preview of nine magnificent papers, and come to know their subject matter much better than I would otherwise have done.

At the centre of this volume is a long paper by Bruce Reed about the tree width of graphs. This is a new measure of connectivity. It is intimately linked to the concept of a minor of a graph, which is obtained by erasing an edge or coalescing two vertices joined by an edge, or by a sequence of such operations.

Summary The group M12 has no transitive extension, but the object of the title is the next best thing: a set of permutations which is an extension of M12. We give an elementary construction, based on a moving-counter puzzle on the projective plane of order 3, and provide easy proofs of some of its properties.

Introduction

Long ago I was intrigued by the fact that M12, É. Mathieu's celebrated quintuply transitive group on 12 letters, shares some structure with L3(3), which acts doubly transitively on the 13 points of the projective plane PG(2,3), of which it is the automorphism group.

To be more precise, the point-stabilizer in L3(3) is a group of structure 32: 2S4 that permutes the 12 remaining points imprimitively in four blocks of 3, and there is an isomorphic subgroup of M12 that permutes the 12 letters in precisely the same fashion. Again, the line-stabilizer in L3(3) is a group of this same structure that permutes the 9 points not on that line in a doubly transitive manner, while the stabilizer of a triple in M12 is an isomorphic group that permutes the 9 letters not in that triple in just the same manner.

In the heady days when new simple groups were being discovered right and left, this common structure inevitably suggested that there should be a new group that contained both M12 and L3(3), various copies of which would intersect in the subgroups mentioned above.

Summary This paper uses an extended example to illustrate how to put together a BDX program to construct block designs fixed by an automorphism, given its orbit matrix. It shows how to specify the parameters and the structural information of the designs. It discusses the symmetry group of the problem and isomorph rejection. It explains how to choose a good order of generation to minimize the size of the search. It also shows how to estimate the size of a search and how to partition the problem into subproblems which can be searched in parallel on several computers.

Introduction

In a recent paper that I co-authored [1], we wrote:

“For each orbit matrix, we used the BDX program to try out all possible circulant matrices with the correct row sum.”

Here, I would like to expand on this sentence, not so much to bore you with details, but to use it as an example to explain how to use the BDX program. This paper is intended as a companion to the BDX reference Guide [7], which is a dry document outlining the syntax and meaning of each of the BDX commands. In this paper, I shall illustrate how the commands can be put together to solve a real problem.

Let me first state what problem we are trying to solve. We want to find all quasi-symmetric 2-(28,12,11) designs with intersection numbers 4 and 6, which are fixed by an automorphism of order 7 without fixed points or fixed blocks.

Summary The harmonious chromatic number of a graph is the least number of colours in a vertex colouring such that each pair of colours appears on at most one edge. The achromatic number of a graph is the greatest number of colours in a vertex colouring such that each pair of colours appears on at least one edge. This paper is a survey of what is known about these two parameters, in particular we look at upper and lower bounds, special classes of graphs and complexity issues.

Introduction

A short survey of harmonious colourings was given by Wilson [80] in 1990. Since then a number of new results have appeared, and the close relationship between harmonious chromatic number and achromatic number has been observed. The purpose of this new survey is to outline what is known about these parameters, and suggest some open problems. A more detailed summary of results on the achromatic number, with a rather different emphasis, can be found in the forthcoming survey by Hughes and MacGillivray [51].

We begin with the definitions of the two parameters.

Definitions A harmonious colouring of a graph G is a proper vertex colouring of G such that, for any pair of colours, there is at most one edge of G whose endpoints are coloured with this pair of colours. The harmonious chromatic number of G, denoted h(G), is the least number of colours in a harmonious colouring of G.

Summary We discuss tree width, a new connectivity invariant of graphs defined by Robertson and Seymour. We present a duality result and a canonical decomposition theorem tied to this invariant. We also discuss a number of applications of these results, including Robertson and Seymour's Graph Minors Project.

Introduction

A taste of things to come

A graph is a set of vertices and an adjacency relation which indicates which pairs of vertices are joined by an edge. Thus, graph theory is essentially the study of connectivity. How then does one measure the connectivity of a graph?

Measuring the connectivity between two vertices is straightforward. Two vertices are said to be k-connected if there are k internally vertex disjoint paths between them. A classical theorem of Menger [30] states that vertices a and b are k-connected in a graph G precisely if there is no set X of fewer than k vertices such that a and b lie in different components of G – X. Standard alternating paths techniques, see e.g. [21], allow us to find either k internally vertex disjoint a-b paths or such a set X efficiently.

An appropriate definition of a highly connected graph, or of a highly connected piece of a graph is more difficult. The classical approach is to call a graph k-connected if every pair of its vertices is k-connected. This definition, although natural, does not capture the kind of connectivity that will concern us. It focuses on local properties rather than global ones.

Summary Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even experimental problems associated with them. In this article, we survey recent work on interval orders and interval graphs, including research on on-line coloring, dimension estimates, fractional parameters, balancing pairs, hamiltonian paths, ramsey theory, extremal problems and tolerance orders. We provide an outline of the arguments for many of these results, especially those which seem to have a wide range of potential applications. Also, we provide short proofs of some of the more classical results on interval orders and interval graphs. Our goal is to provide fresh insights into the current status of research in this area while suggesting new perspectives and directions for the future.

Introduction

A complex process (manufacturing computer chips, for example) is often broken into a series of tasks, each with a specified starting and ending time. Task A precedes Task B if A ends before B begins. When A precedes B, the output of A can safely be used as input to B, and resources dedicated to the completion of A, such as machines or personnel, can now be applied to B. When A and B have overlapping time periods, they may be viewed as conflicting tasks, in the sense that they compete for limited resources.

Summary A permutation group on a set Ω is said to be quasiprimitive on Ω if each of its nontrivial normal subgroups is transitive on Ω. For certain families of finite arc-transitive graphs, those members possessing subgroups of automorphisms which are quasiprimitive on vertices play a key role. The manner in which the quasiprimitive examples arise, together with their structure, is described.

Introduction

A permutation group on a set Ω, is said to be quasiprimitive on Ω if each of its nontrivial normal subgroups is transitive on Ω. This is an essay about families of finite arc-transitive graphs which have group-theoretic defining properties. By a quasiprimitive graph in such a family we shall mean a graph which admits a subgroup of automorphisms which not only is quasiprimitive on vertices, but also has the defining property of the family. For example, in the family of all arc-transitive graphs, a quasiprimitive graph is one with a subgroup of automorphisms which is both quasiprimitive on vertices and transitive on arcs. (An arc of a graph Γ is an ordered pair of adjacent vertices.)

First we shall describe an approach to studying several families of finite arctransitive graphs whereby quasiprimitive graphs arise naturally. The concept of quasiprimitivity is a weaker notion than that of primitivity for permutation groups, and we shall see that finite quasiprimitive permutation groups may be described in a manner analogous to the description of finite primitive permutation groups provided by the famous theorem of M. E. O'Nan and L. L. Scott [17, 30]. There are several distinct types of finite quasiprimitive permutation groups, and several corresponding distinct types of finite quasiprimitive graphs.