Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs
Keywords:
Graph theory, Matroid Theory, Gammoids, Infinite graph, Ends, Topological infinite graph theory, Menger's theorem
Abstract
Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.
As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.
This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.