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.