This book contains articles based on talks at the Second International Conference on Finite Geometry and Combinatorics, which took place from 31 May to 6 June 1992, at the Conference center De Ceder in Astene–Deinze, Belgium. There were 76 participants and 52 talks.

There are 35 articles in these proceedings.

The editors, who were also the conference organisers, are grateful for the financial support of the National Fund for Scientific Research (NFWO) of Belgium and of the University of Ghent.

Above all we are grateful to Zita Oost, the conference secretary, and to Ruth Lauwaert, who helped a great deal with the production of the copy.

Abstract

A generalized Fischer space is a partial linear space in which any two intersecting lines generate a subspace isomorphic to an affine plane or the dual of an affine plane. We give a classification of all finite and infinite generalized Fischer spaces under some nondegeneracy conditions.

Introduction

Let Π = (P, L) be a partial linear space, that is a set of points P together with a set L of subsets of P of cardinality at least 2 called lines, such that each pair of points is in at most one line. A subset X of P is called a subspace of Π if it has the property that any line meeting X in at least two points is contained in X. A subspace X together with the lines contained in it is a partial linear space. Subspaces are usualy identified with these partial linear spaces. As the intersection of any collection of subspaces is again a subspace, we can define for each subset X of P the subspace generated by X to be the smallest subspace containing X. This subspace will be denoted by 〈X〉. A plane is a subspace generated by two intersecting lines.

In [3], [6] partial linear spaces are considered in which all planes are either isomorphic to an affine plane or the dual of an affine plane. Such spaces are called generalized Fischer spaces.

Abstract

We completely determine the subgroups, which also are subplanes, of a Singer group of planar order 81. We prove that each subgroup of a Singer group is invariant under the involution of the multiplier group, except possibly if the Singer group is non abelian of planar order 16. If the subgroup is a subplane of non square order, then this subplane is centralized by the involution of the multiplier group. We study v(n) = v(x)v(y)v(z) from a geometrical point of view, where n is the order of a projective plane and v(r) = r2 + r + 1 for any r.

Introduction

A Singer group of a projective plane is a collineation group acting regularly on the points of the plane. In 1938, Singer proved that a finite Desarguesian plane admits a cyclic Singer group. On the other hand, in 1964, Karzel proved that a plane admitting an infinite cyclic Singer group is not Desarguesian. Projective planes and Singer groups in this article are of finite cardinalities. An automorphism of a Singer group is a multiplier if it is also a collineation when we identify the points of the plane with the elements of the group. The set of all multipliers is called the multiplier group of the Singer group.

Discrete mathematics has had many practical applications in recent years and this is only one of the reasons for its increasing dynamism. The study of finite structures is a broad area which has a unity not merely of description but also in practice, since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics, and we are happy to be able to demonstrate progress.

It was a great pleasure to see several Russian colleagues participating both because they were able to do so, some for the first time, and because this is an area of Mathematics not as diffuse in Russia as elsewhere. It was also good to see the participation of a significant number of talented, younger colleagues, but at the same time sad to note the difficulty they are having in finding permanent positions.

The conference papers are here divided into themes. The division is somewhat artificial as some papers could be placed in more than one group. The style of mathematics is very much resolving problems rather than the construction of grand theories. There are still many puzzling features about the sub-structures of finite projective spaces, as well as about finite strongly regular graphs, finite projective planes, and other particular finite diagram geometries. Finite groups are as ever a strong theme for several reasons.

Basics of complexity

The basic notions of computational complexity are now familiar concepts in most branches of mathematics. One of the main purposes of the theory is to separate tractable problems from the apparently intractable. Deciding whether or not P = NP is a fundamental problem in theoretical computer science. We will give a brief informal review of the main concepts.

We regard a computational problem as a function, mapping inputs to solutions, (graphs to the number of their 3-vertex colourings for example). A function is polynomial time computable if there exists an algorithm which computes the function in a length of time (number of steps) bounded by a polynomial in the size of the problem instance. The class of such functions we denote by FP. If A and B are two problems we say that A is polynomial time Taring reducible to B, written A ∞ B, if it is possible with the aid of a subroutine for problem B to solve A in polynomial time, in other words the number of steps needed to solve A (apart from calls to the subroutine for B) is polynomially bounded.

The difference between the widely used P and the class FP is that, strictly speaking, both P and NP refer to decision problems.

A typical member of NP is the following classical problem known as SATISFIABILITY, and often abbreviated to SAT.

These lecture notes axe based on a series of lectures which I gave at the Advanced Research Institute of Discrete Applied Mathematics (ARIDAM VI) in June 1991.

The lectures were addressed to an audience of discrete mathematicians and computer scientists. I have tried to make the material understandable to both groups; the result is that there are introductions to topics such as the complexity of enumeration, knots, the Whitney/Tutte polynomials and various models of statistical physics.

The main thrust throughout is towards algorithms, applications and the interrelationship among seemingly diverse problem areas. In many cases I have only given sketches of the main ideas rather than full proofs. However, I have tried to give detailed references. I have assumed some familiarity with the basic concepts of computational complexity and combinatorics, but I have aimed to define anything nonstandard when it is first encountered. My notation in both cases corresponds to standard usage, such as Garey and Johnson (1979) and Bollobás (1979).

Since the lectures I have rewritten the notes to incorporate some of the new developments but the basic material is the essence of what was presented. Much of the work was done when I held a John von Neumann Professorship at the University of Bonn. I am very grateful for the opportunity this offered, and to my friends at the Forschungsinstitut für Diskrete Mathematik, where the facilities and atmosphere make it such a stimulating place to visit.