Arboricity and tree-packing in locally finite graphs

Abstract

Nash-Williams' arboricity theorem states that a finite graph is the edge-disjoint union of at most k forests if no set of ℓ vertices induces more than k(ℓ - 1) edges. We prove a natural topological extension of this for locally finite infinite graphs, in which the partitioning forests are acyclic in the stronger sense that their Freudenthal compactification - the space obtained by adding their ends - contains no homeomorphic image of S1. The strengthening we prove, which requires an upper bound on the end degrees of the graph, confirms a conjecture of Diestel [The cycle space of an infinite graph, Combin. Probab. Comput. 14 (2005) 59-79]. We further prove for locally finite graphs a topological version of the tree-packing theorem of Nash-Williams and Tutte. © 2005 Elsevier Inc. All rights reserved.