A Proof of the Bounded Graph Conjecture

Let σ, τ : N→N be two sequences of natural numbers. We say that σ dominates τ if σ(n) τ(n) for every n greater than some n0 ∈ N. Now let G be an infinite graph, and assume the vertices of G are labelled with natural numbers, say by f : V (G)→N. Then each ray (one-way infinite path) R = x0x1 . . . in G gives rise to a sequence τR := (f(x0), f(x1), . . .); we say that σ dominates R if σ dominates this sequence τR. We may then ask whether or not there exists a sequence σ: N → N which dominates all the rays in G simultaneously. If this is the case, we say that the labelling f of G is bounded by σ; if not, f is unbounded . An unlabelled graph G is said to be bounded if every labelling of its vertices is bounded. Thus, G is bounded if and only if for every labelling f : V (G)→N there exists a sequence σ: N→N which dominates every ray in G with respect to f . If not, G will be called unbounded . Our main aim in this paper is to give a proof of what has become known as the ‘bounded graph conjecture’ [ 5 ]. This conjecture, proposed by Halin in 1964 (see [ 5 ]) but first published in [ 4 ], characterizes the bounded graphs by the exclusion of four simple prototypes of unbounded graphs as topological subgraphs. Not surprisingly, this characterization has some fundamental implications for the concept of boundedness.