On spanning trees and k-connectedness in infinite graphs

If two rays P,Q ⊂ G satisfy (i)–(iii), we call them end-equivalent in G. An end of G is an equivalence class under this relation, and E(G) denotes the set of ends of G. For example, the 2-way infinite ladder has two ends, the infinite grid Z×Z and every infinite complete graph have one end, and the dyadic tree has 20 ends. This paper is concerned with the relationship between the ends of a connected graph G and the ends of its spanning trees. If T is a spanning tree of G and P,Q are end-equivalent rays in T , then clearly P and Q are also endequivalent in G. We therefore have a natural map η : E(T )→E(G) mapping each end of T to the end of G containing it. In general, η need be neither 1–1 nor onto. For example, the 2-way infinite ladder has a spanning tree with 4 ends (the tree consisting of its two sides together with one rung), and every infinite complete graph is spanned by a star, which has no ends at all. A spanning tree T of G for which η is 1–1 and onto is called end-faithful . The concept of ends in graphs, and of end-faithful spanning trees, was introduced by Halin [ 4 ] in 1964. Halin asked whether every infinite connected graph has an end-faithful spanning tree, and proved that this is so for all countable graphs. End-faithful spanning trees have since been constructed for some classes of uncountable graphs as well (see [ 2 ] and, especially, Polat [ 8 ]), but very recent results due to Seymour and Thomas [ 10 ] and to Thomassen [ 12 ] show that some uncountable graphs have no such tree. See [ 3 ] for an up-todate survey of results and open problems in this field.