I'm looking at a proof that every connected graph G contains a maximal tree T (that is, a tree such that for any edge e in G, T(union)e is no longer a tree).
The proof begins by enumerating the (possibly countably infinite) vertices of G.
The proof then asserts that we may assume that for any i>=2, the ith vertex shares an edge with a jth vertex where j<i.
This seems intuitive but I can't see exactly how we could ensure that this is the case.
Any help with seeing why we may assume this would be a great help. Thanks!