A simple existence criterion for normal spanning trees in infinite graphs

A spanning tree of an infinite graph is normal if the endvertices of any chord are comparable in the tree order defined by some arbitrarily chosen root. (In finite graphs, these are their ‘depth-first search’ trees; see [3] for precise definitions.) Normal spanning trees are perhaps the most important single structural tool for analysing an infinite graph (see [4] for a good example), but they do not always exist. The question of which graphs have normal spanning trees thus is an important question. All countable connected graphs have normal spanning trees [3]. But not all connected graphs do. For example, if T is a normal spanning tree of G and G is complete, then T defines a chain on its vertex set. Hence T must be a single path or ray, and G is countable. For connected graphs of arbitrary order, there are three characterizations of the graphs that admit a normal spanning tree: