A proof of the bounded graph conjecture.(English)Zbl 0793.05121

Given a ray $$R: v_ 1 v_ 2\dots$$ in the graph $$G$$ and a function $$f: V(G)\to\mathbb{N}$$, the sequence $$\sigma: V(G)\to\mathbb{N}$$ is said to dominate $$f$$ on $$R$$, if $$\sigma(n)\geq f(v_ n)$$ for all but finitely many $$n\in\mathbb{N}$$. The graph $$G$$ is called bounded if for each $$f: V(G)\to\mathbb{N}$$ some sequence $$\sigma$$ dominates $$f$$ on every ray in $$G$$. The authors characterize the bounded countable graphs as those not containing a subdivision of any of three particular countable graphs as subgraphs. This characterization was conjectured by Halin in 1964. General bounded graphs $$G$$ are characterized by the additional property that $$G$$ must not contain $$\kappa$$ disjoint rays, where $$\kappa$$ is a certain cardinal such that $$\omega<\kappa\leq 2^ \omega$$ ($$\kappa= \omega_ 1$$ if $$2^ \omega= \omega_ 1$$). Also the exclusion of the three (four) basic graphs as minors characterizes the countable (general) bounded graphs.
Reviewer: H.A.Jung (Berlin)

MSC:

 05C99 Graph theory 05C10 Planar graphs; geometric and topological aspects of graph theory 05C75 Structural characterization of families of graphs
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References:

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