Hi

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!