Arboricity and tree-packing in locally finite graphs

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.