Introduction

Let WFGn be the class of graphs on n vertices with the property that the neighborhood of any vertex is acyclic. A graph G is given by its vertex set V(G) and edge set E(G). The subgraph induced by X ⊂ V(G) is denoted by G[X]. The neighborhood N(v) of vertex v is the set of vertices adjacent to v. Note that v ∉ N(v). The degree of v ∈ V(G), denoted by dv or dv(G), is the size of the neighborhood: dv = |N(v)|. The maximum (minimum) degree is denoted by Δ (δ), or Δ(G) (δ(G), respectively) to avoid misunderstandings. A matching M ⊂ E(G) is a set of pairwise disjoint edges. A wheel Wi is obtained from a cycle Ci by adding a new vertex and edges joining it to all the vertices of the cycle; the new edges are called the spokes of the wheel (i ≥ 3, W3 = K4). Therefore, WFGn consists of all graphs on n vertices containing no wheel.

Introduction

In this paper we will characterize those general graphs with degrees 2k − 2 and 2k that can be decomposed into spanning subgraphs with degrees 1 and 2 everywhere. Before we state the result, it is perhaps of some interest to review some related problems and their history.

Background

One of the starting points of graph theory is a classic investigation by the Danish mathematician Julius Petersen who in 1891 published a paper: ‘Die Theorie der regulären graphs’, which contains a wealth of material on the problem of factorizing regular graphs into graphs of uniform degree k. An excellent source of information concerning Julius Petersen and problems spawned by his 1891 paper is the conference volume.

The motivation for Petersen's work, as given in the first few lines of his article, came from Hilbert's proof of the finiteness of the system of invariants associated with a binary form.

Introduction

We consider undirected graphs without loops or multiple edges. The graph Kn on n vertices for which every pair of distinct vertices induces an edge is called a complete graph or a clique on n vertices. If G is any graph, we call any complete subgraph of G a clique of G (we do not require that it be a maximal complete subgraph). A clique covering of G is a set of cliques of G that together contain each edge of G at least once; if each edge is covered exactly once we call it a clique partition. The clique covering number cc(G) and clique partition number cp(G) are the smallest cardinalities of, respectively, a clique covering and a clique partition of G.

The question of calculating these numbers was raised by Orlin in 1977. DeBruijn and Erdős had already proved, in 1948, that partitioning Kn into smaller cliques required at least n cliques. Some more recent studies motivating the current paper include.

A hypergraph H on a set of points V is simply a collection of subsets E of V, the edges of H. A d-graph is a hypergraph in which each edge has size d. A weak 2-colouring of a hypergraph is a partition of the points into two ‘colour’ sets A and B such that each edge E meets both A and B. Deciding if a 3-graph has a weak 2-colouring is NP-complete.

The following simple randomised recolouring method attempts to find a weak 2-colouring of a hypergraph H. It is assumed that we have a subroutine SEEK, which on input of a 2-colouring of the points outputs a monochromatic edge if there is one, and otherwise reports that there are none.

Introduction

Let K be a subgraph of G. A K-component or a K-bridge in G is a subgraph of G that is either an edge e ∈ E(G)\E(K) (together with its endpoints) that has both endpoints in K, or it is a connected component of G − V(K) together with all edges (and their endpoints) between this component and K. Each edge of a K-component R having an endpoint in K is a foot of R. The vertices of R ∩ K are the vertices of attachment of R. A vertex of K of degree in K different from 2 is a main vertex of K.

Introduction

For many mathematicians the most noble activity lies in proving theorems. It must have come as a blow for them when Gödel [7] showed that there are unprovable theorems. At the beginning they still could find some consolation in hoping that such culprits might only occur in Peano arithmetics through esoteric diagonalization arguments. Nowadays there is a wealth of the most natural valid theorems that can be stated in the language of finite combinatorics but are not provable within that system.

Mathematicians understand to a certain extent how to find unprovable theorems and how to prove their unprovability within a formal system. In that sense we are relying on the classical work by Gentzen [5], Kreisel [15] and Wainer [31]. Moreover, we shall apply their beautiful ideas to something that seems to be well understood, viz to well-quasi-orderings. This is an old concept found in Gordan [6], and Kruskal [16] correctly pointed out that it was ‘a frequently discovered concept’. That is why we are not reinventing it and are well aware that any sequence (si) of specialists starting with the author must contain an arbitrary long subsequence of experts knowing more than s0, a fact, which gives a nice theme for this paper.

Introduction

How well can global properties of a graph be inferred from observations that are purely local? This general question gives rise to numerous interesting problems that we want to discuss here. Such a local-global approach is often taken in geometry, where it has a long and successful history, but a systematic study of graphs from this perspective has not begun until recently. Nevertheless, a number of older results in graph theory do fit very nicely into this framework, as we later point out. Most of the specific problems fall in two categories. In the first, local structural information on the graph is collected and then used to derive certain consequences for the graph as a whole. The other class of problems concerns consistency of local data. Namely, one asks to characterize those sets of local data that may come from some graphs.

As the reader will soon see, the local-global paradigm leads to many questions in which graphs are viewed as geometric objects, a point of view that we believe can greatly benefit graph theory. Besides the geometric connection, ties also exist with the theory of combinatorial algorithms.

Introduction

We shall be concerned with a graph G regarded as an electrical network in which each edge has resistance 1. A well-known result due to R.M. Foster [6] (see also [3, p.41] and [9]) asserts that if G has n vertices and m edges, the effective resistance between adjacent vertices is r1 = (n − 1)/m, provided that all edges are equivalent under the action of the automorphism group. In this paper I shall obtain formulae for ri, the effective resistance between vertices at distance i, for i ≥ 2, provided G is distance-transitive (DT). With hindsight, it will be clear that the same formulae hold if we assume only that G is distance-regular (DR). The case i = 2 was also studied by Foster [7].

Another well-known fact is that the electrical problem can be regarded as a case of solving Laplace's equation on the graph.