An elementary proof of Frank’s constructive characterization of the graphs having k edge disjoint spanning trees

Abstract

We give an elementary proof of Frank’s Theorem stating that a (finite, undirected, nonempty) multigraph has $k$ edge disjoint spanning trees if and only if it can be obtained from $K_1$ by repeatedly (i) adding an edge or (ii) chosing a sequence $\sigma$ of $k$ vertices and pairwise distinct non-loop edges, deleting the edges of $\sigma$, and adding a new vertex plus one edge to each vertex of $\sigma$ plus one edge to each end of every edge of $\sigma$.