Kriesell's conjecture for infinite graphs

Abstract

Let $G$ be a graph and $S\subseteq V(G)$ be a subset of vertices. An $S$-Steiner tree $T$ of $G$ is a tree of $G$ which contains $S$ in its vertex set $V(T)$. Kriesell conjectured that for every $2k$-edge-connected subset $S\subseteq V(G)$ in a finite connected graph $G$, there exist $k$ pairwise edge-disjoint $S$-Steiner trees. This conjecture is false for infinite graphs. We present a version of Kriesell's conjecture with topological $S$-Steiner trees for countable finitely edge-separable graphs and a version with $F$-limits of trees for rayless graphs. We show that if Kriesell's conjecture holds for finite graphs, then it holds for every connected, rayless and finitely edge-separable graph. We also show that every $2k$-edge-connected rayless and finitely edge-separable graph contains $k$ pairwise edge-disjoint spanning trees.