Spanning trees in hyperbolic graphs

Abstract

We construct spanning trees in locally finite hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has at least one but at most a bounded number of disjoint rays to each boundary point. As a corollary we extend a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree (disjoint from the graph) with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We shall construct a tree with these properties as a subgraph of the hyperbolic graph, which in addition is also a spanning tree of that graph.

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Correspondence to Matthias Hamann.

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Hamann, M. Spanning trees in hyperbolic graphs. Combinatorica 36, 313–332 (2016). https://doi.org/10.1007/s00493-015-3082-2

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Mathematics Subject Classification (2010)

  • 05C05
  • 20F67
  • 05C63