On October 29, 2019, Pascal Gollin from IBS discrete mathematics group gave a talk on the existence of a decomposition of an infinite graph into spanning trees in terms of the existence of packing and covering of spanning trees at the discrete math seminar. The title of his talk was “A Cantor-Bernstein-type theorem for spanning trees in infinite graphs“.
Given a cardinal $\lambda$, a $\lambda$-packing of a graph $G$ is a family of $\lambda$ many edge-disjoint spanning trees of $G$, and a $\lambda$-covering of $G$ is a family of spanning trees covering $E(G)$.
We show that if a graph admits a $\lambda$-packing and a $\lambda$-covering then the graph also admits a decomposition into $\lambda$ many spanning trees. In this talk, we concentrate on the case of $\lambda$ being an infinite cardinal. Moreover, we will provide a new and simple proof for a theorem of Laviolette characterising the existence of a $\lambda$-packing, as well as for a theorem of Erdős and Hajnal characterising the existence of a $\lambda$-covering.
Joint work with Joshua Erde, Attila Joó, Paul Knappe and Max Pitz.