By a graph G we mean here a linear network, at least 3-connected with no vertices of degree 2, no multiple edges and no loops. A graph T is called a spanning graph of G provided (i) T is a tree, (ii) T is a subgraph of G and (iii) every vertex of G belongs to T.

A graph is said to be homeomorphically irreducible (HI) if it has no nodes of degree 2. Thus an HI tree may be a spanning tree of G and we shall call this graph a HISTree of G.

Starting with the example of the set of all 3-polytopes, inspection shows that while all possess spanning trees [1] not all have HISTrees.(1) We would like to know which graphs possess HISTrees, if such graphs are common and what may be found as contingent with the existence or non-existence of this feature.

Types of HISTrees

For the purpose of the present note it is sufficient to regard these graphs as belonging to three main sub-species. These are:

1. (i) Stars (as in fig. 4, which shows the smallest).

2. (ii) Star chains (as in figs. 5, 6. 1 and 10. 1).

3. (iii) Star trees (as in fig. 7.1).

Types of 3-polytopes with HISTrees

The most obvious example of a family of graphs G containing a HISTree is the family of those 3-polytopes in which those edges not belonging to the HISTree comprise a circuit joining the terminal vertices of the HISTree.

Introduction

Let G, H be two finite graphs (possibly with loops but with no multiple edges) having vertex sets V(G), V(H) and edge sets E(G), E(H). An edge joining vertices u and v is denoted by [u, v] and, therefore, a loop at u by [u, u], A function f : V(G) → V(H) is called a homomorphism if it preserves adjacencies, i. e. if [u, v] ∈ E(G) implies [f(u), f(v)] ∈ E(H). The set of all such homomorphisms is denoted by hom(G, H). The set of all one-one homomorphisms from G to itself forms the automorphism group of G denoted by aut(G). It is often desirable to regard f, g ∈ hom(G, H) as equivalent if there exists a ∈ aut(G) such that f = ga. The set of equivalence classes is denoted by hom(G, H)/aut(G).

A colouring of a graph G is a function k : V(G) → C where C is a finite set whose elements are called colours. The following general problem is of some interest: how many colourings of G are there subjec to certain specified adjacency restrictions - i. e. as well as C a matrix B is given with (B)ij = 1 (or 0) if colour i may (or may not) be adjacent to colour j. Of course, B may be regarded as the adjacency matrix of a graph H. Each colouring of G (subject to the restrictions) corresponds to an element of hom(G, H) and conversely.

In statistical mechanics we are interested in various problems of counting subgraphs of a periodic lattice and I shall consider the plane square lattice as an illustration. Virtually all the analytically solved problems on this lattice either involve the counting of subgraphs all of whose vertices are of even valency, or can be transformed into such problems [3]. I consider a problem to be ‘analytically solved’ if we know the limiting form of the generating function in the limiting case when the numbers and lengths of the rows become very large.

One of the main methods of handling these problems is by a matrix method [3]. The matrix describes the operation of adding a row of points to the lattice. Each row of the matrix corresponded to one of the possible configurations of vertical lines in row m of the lattice, each column of the matrix to one of the possible configurations of vertical lines in row m + 1 of the lattice. Configurations of the whole lattice are then enumerated by a high power of this matrix and the crucial problem is to determine its largest eigenvalue. For a row of n points this is a 2n × 2n matrix.

I have no brand new results to report, but I think that it will be useful to describe a method that has been used for quite a number of these problems. It was originally used by Bethe for the rather different problem of the magnetisation of a one-dimensional lattice.

The British Combinatorial Conference, 2nd-6th July 1973, held at Aberystwyth, was the fourth such conference held in Britain in the past decade.

Most lectures were of twenty to forty minutes duration. There were, however, four longer lectures from invited speakers. These four were H. Lüneburg (Kaiserslautern), C. St. J. A. Nash-Williams (Aberdeen), W. T. Tutte (Waterloo) and J, H. van Lint (Eindhoven).

It was envisaged that this conference should be one of a regular series, aimed at providing a forum for combinatorial mathematicians in Britain. It was also hoped that there would be considerable participation by mathematicians from outside Britain. That this hope was fulfilled is seen from the substantial number (thirty per cent) of such mathematicians attending.

We would like to thank the University College of Wales, and in particular its Department of Pure Mathematics, for the assistance given to the organisers both prior to, and during, the conference. We also wish to acknowledge, with gratitude, the financial assistance given by the Royal Society.

Introduction

The well-known theorem of H. J. Ryser [12] giving necessary and sufficient conditions for an r × s latin rectangle on 1, …, n to be embedded in an n × n latin square on 1, …, n was used by T. Evans [2] (and independently by S. K. Stein) to show that an incomplete n × n latin square on 1, …, n can be completed to a 2n × 2n latin square on 1, …, 2n. A similar, but somewhat more complicated, pair of theorems concerning symmetric latin squares was proved by A. Cruse [1].

The purpose of this note is to give alternative and, in my opinion, simpler proofs of the theorem of Ryser and the analogous theorem of Cruse. Ryser's theorem generalizes M. Hall's theorem [31 that an r × n latin rectangle on 1, …, n can be embedded in an n × n latin square on 1, …, n, but the methods of proof seem to be rather dissimilar. The proof of RyserTs theorem which is given here is very obviously a simple generalization of the original proof of M. Hall's theorem.

There are still some open problems in this area (see [5], [71), so it is possible that the existence of these alternative proofs may help towards the solution of some of these problems.