Maps and surfaces

In this chapter we shall discuss graphs realized by a set of points and lines on a closed orientable surface. Although this notion arises in a topological context, we shall show that it is possible to develop it by purely combinatorial arguments. The reader who is unfamiliar with the topology of surfaces should not be at a disadvantage.

We begin at an intuitive level. For our purposes, it is sufficient to say that a ‘surface’ is a compact topological space which has two special properties:

1. (i) it is locally homeomorphic to ordinary Euclidean 2-space;

2. (ii) it has a consistent global orientation.

The sphere and the torus are the simplest examples. The Euclidean plane is not compact, and so it is not a ‘surface’ for us; however, it may be made homeomorphic to a sphere by the addition of a single point, and so the two spaces have very similar properties. The Klein bottle is not allowed, since it has no consistent global orientation.

Let us suppose that a graph Γ is represented by a set of points and lines on a surface, in such a way that the lines intersect only at the points representing their end vertices.

INTRODUCTION

It is only too easy for the proliferation of results involving symmetry conditions in graphs to conform to no very obvious pattern. I should like to try to impose a simple conceptual framework on those sections of the subject in which I have been chiefly involved, by considering a sequence of examples which seem to me to emerge in a natural way. The treatment is by no means either uniform or exhaustive – emphasis and detail reflecting my own personal interests.

All graphs Γ will be finite, undirected, without loops or multiple edges, and are generally assumed to be connected. (Though the ideas usually extend in one way or another to directed graphs, the results and methods of proof rarely do.)

The framework I shall propose arises by first exploring the most extreme symmetry condition one might use to define a class of graphs; afterwards we consider the ways in which such extremism might be most profitably modified.

EXAMPLE I

A graph Γ is (globally) homogeneous if whenever U,V ⊆ VΓ gives rise to isomorphic induced subgraphs <U>,<V>, every isomorphism σ: <U> → <V> extends to an isomorphism of Γ.

Here no restriction is placed on the isomorphism type of the induced subgraphs <U>,<V>, and no assumption is made about the way U and V are embedded in Γ. The assumption is distressingly strong, and one is not surprised to find a complete classification of such graphs.

SUMMARY

This article is another attempt to reconcile part of graph theory and part of theoretical physics. Specifically, we shall discuss some aspects of the reconstruction problem in terms of simple models of physical phenomena. A previous essay in the same vein (Biggs 1977a, henceforth referred to as IM) may be consulted for background information and proofs of some basic theorems.

There are four sections: (1) Interaction Models, (2) the Algebra of Graph Types, (3) Reconstruction, (4) Partition Functions for Infinite Graphs.

INTERACTION MODELS

Definitions

Let G be a finite (simple) graph, with vertex set V and edge-set E. An ‘interaction model’ on G arises when the vertices of G may have certain attributes, and they interact along the edges according to the values of those attributes. To be more precise, let A denote a finite set of objects (attributes), and define a state on G to be a function ω: V → A. In graph theory, we often think of the attributes as colours, so that a state is a colouring. In theoretical physics, the vertices represent particles of some kind, and the attributes represent some physical property, such as a magnetic moment.

The interaction between a pair of adjacent vertices is measured by a real-valued function i, called the interaction function, defined on the set A(2) of unordered pairs of attributes. That is, i{a1,a2) measures the interaction between a pair of vertices when they are joined by an edge and they have attributes a1 and a2. An interaction model M is a pair (A,i), where A is a finite set and i is an interaction function.

This talk is a review of recent developments in the study of extremal properties of set systems or hypergraphs.

We will begin with some examples of classical problems in this area; we will then discuss some recent results, and will finally discuss several conjectures and open problems. We apologize in advance for slighting important work (or contributors); in defence we note that while a review article can try to be encyclopaedic, nobody should be asked to listen to an encyclopaedic talk.

A graph is generally defined as a set of vertices and a set of pairs of vertices called arcs or lines or edges. A hypergraph similarly has vertices and hyperedges, which latter are here sets of vertices. A hypergraph is therefore another name for a collection of subsets of the vertices. We will refer particularly to several kinds of hypergraphs. A k-hypergraph has each of its members consisting of k vertices; an intersecting hypergraph has every pair of edges having at least one vertex in common. An antichain has no edge containing another; and an ideal has the property that any set of vertices contained in an edge is also an edge.

An extremal property is a statement that whatever possesses that property has to be or cannot be bigger than something or other.

The area that we are concerned with is not really extremal properties of hypergraphs per se (these are staggeringly trivial) but rather extremal properties of kinds of hypergraphs. Of course almost every mathematical construction contains within it some sort of hypergraph – so that almost any extremal property relates somehow to a kind of hypergraph.

INTRODUCTION

The present paper is meant to give a selfcontained introduction to the theory (rather than to the construction) of strongly regular graphs, with an emphasis on the configurational aspects and the Krein parameters.

The definition in graph-theoretical terms in §2 is translated into the language of matrices and eigenvalues in §3, leading to a short arithmetic of the parameters in §4. In §5 the adjacency algebra relates the combinatorial and the algebraic idempotents, leading to spherical 2-distance sets and the Krein inequalities. The absolute bound of §6 provides inequalities of a different type. Necessary and sufficient conditions for equality in either bound are considered in §7, as well as their graph-theoretic significance. Finally, the notion of switching is explained in §8, and illustrated by two explicit examples in §9.

Strongly regular graphs have been introduced by Bose [1], related to algebra by Bose and Mesner [2], to eigenvalues by Hoffman [11], and to permutation groups by D.G. Higman [10]. Scott [17] introduced the Krein parameters. Margaret Smith [20] considered extremal rank 3 graphs, providing a motivation for the recent work by Cameron et al. [7], whose pattern was followed in the present paper. For a survey of construction methods for strongly regular graphs the reader is referred to Hubaut [12], and for relations with other combinatorial objects to Cameron – van Lint [5], Cameron [6] and Goethals-Seidel [8],[9].

DEFINITIONS

We shall deal with ordinary graphs (undirected, without loops and multiple edges) having a finite number n of vertices. For any pair of vertices a between-vertex is a vertex adjacent to both.

INTRODUCTION

There is an extensive literature on long cycles, in particular Hamiltonian cycles, in undirected graphs. Since an undirected graph may be thought of as a symmetric digraph it seems natural to generalize (some of) these results to digraphs. However, cycles in digraphs are more difficult to deal with than cycles in undirected graphs and there are relatively few results in this area. In the present paper we review the results on long cycles, in particular Hamiltonian cycles, in digraphs with constraints on the degrees, and we present a number of unsolved problems.

TERMINOLOGY

We use standard terminology with a few modifications explained below. A digraph (directed graph) consists of a finite set of vertices and a set of ordered pairs xy of vertices called edges. If the edge xy is present, we say that x dominates y. The outdegree d+ (x) of x is the number of vertices dominated by x, the indegree d− (x) is the number of vertices dominating x, and the degree d(x) of x is defined as d+ (x) + d− (x).

A digraph D is k-regular if each vertex has degree k and D is k-diregular if each vertex has indegree k and outdegree k. The order of a digraph is the number of vertices in it. When we speak of paths or cycles in digraphs, we always mean directed paths or cycles. A k-cycle is a cycle of length k. A digraph of order n is pancyclic if it contains a k-cycle for each k = 2,3,…,n.

INTRODUCTION

One of the attractions of matroid theory is that it usually provides several different natural settings in which to view a problem. The theme of this paper is the use of matroids to relate problems about colourings and flows in graphs with problems of projective geometry.

Veblen in 1912 was the first to attempt to settle the four colour conjecture by geometrical methods. In the same year G.D. Birkhoff tried to settle it by an enumerative approach. Despite, or perhaps because of, their lack of success their work led to some startling extensions, notably by W.T. Tutte. Following on from Veblen's work, Tutte in 1966 formulated a fascinating geometrical conjecture – namely that there were just three minimal geometrical configurations which had a non empty intersection with each coline of PG(n,2) (he called them tangential 2-blocks). As a result of his work on enumerating polynomials Tutte in 1954 was led to make a series of equally fascinating conjectures about flows in directed networks. One of these was recently proved by F. Jaeger in 1975, when he gave a very elegant proof that every bridgeless graph has an 8-flow – equivalently every bridgeless graph is the union of three Eulerian subgraphs. Even more recently a remarkable advance towards the solution of Tutte's tangential block conjecture was made by P.D. Seymour when he showed that the only new tangential blocks must be cocycle matroids of graphs. This reduces a seemingly intractable geometrical question to the conceptually easier problem of classifying those graphs which cannot be covered by two Eulerian subgraphs.

In the last twelve years more and more combinatorialists have taken interest in connectivity problems, and therefore some progress has been made, but there are still more unsolved problems than solved ones. We shall confine ourselves here to finite, undirected graphs and only sometimes we shall mention analogous problems for infinite graphs or for digraphs. Most of the connectivity problems for undirected graphs have a counterpart in the directed case. In general, these “directed” problems are more complicated, and so it may happen that a connectivity problem is completely solved for undirected graphs whereas the corresponding problem for digraphs has not even been attacked. (But there are some connectivity problems for digraphs which have no analogue for undirected graphs, as for instance the question answered in [22], Also the intermediate result of Nash-Williams [54] (cf. also [45]) should be mentioned.) If not otherwise stated, all graphs are supposed to be undirected and finite without multiple edges or loops.

The article is divided into four parts. In part I we determine the maximum number of openly disjoint or edge-disjoint paths joining vertices of a given set. In part II we try to construct all n-connected (n-edge-connected) graphs and collect some properties of n-connected graphs. In part III we look for some special configurations (circuits through given vertices or edges, subdivisions of complete graphs) in n connected graphs and in part IV we ask for the maximum number of edges a graph with m vertices may have without containing such configurations as vertices x ≠ y joined by n openly disjoint (or edge-disjoint) paths or an n-connected (or n-edge-connected) subgraph.

Since its inception at Oxford in 1969 the British Combinatorial Conference has become a regular feature of the international mathematical calendar. This year the seventh conference will be held in Cambridge from 13th to 17th August, under the auspices of the Department of Pure Mathematics and Mathematical Statistics. The participants and the contributors represent a large variety of nationalities and interests.

The principal speakers were drawn from the mathematicians of Britain, Europe and America. They were asked to review the diverse areas of combinatorics in which they are expert. In this way it was hoped to provide a valuable work of reference describing the state of the art of combinatorics. All of the speakers kindly submitted their articles in advance enabling them to be published in this volume and made available in time for the conference.

I am grateful to the contributors for their cooperation which has made my task as an editor an easy one. I am also grateful to the Cambridge University Press, especially Mr David Tranah, for their efficiency and skill. On behalf of the British Combinatorial Committee I would like to thank the British Council, the London Mathematical Society and the Mathematics Faculty of Cambridge University for their financial support.