A note on Menger's theorem for infinite locally finite graphs

Abstract

Introduction. In [1] the ends of an infinite graph G were introduced; they are the equivalence classes induced by the following equivalence relation on the set ~1 (G) of all one way infinite paths in G: Set U ~V for U, V e ~ I ( G ) if and only if there is a W e ~ I ( G ) meeting both U and V in an infinite number of vertices. The ends of G may be considered as something like improper or figurative vertices of G; the intui t ion behind this consideration of ends as figurative vertices is somewhat related to the idea of the projective closure of an affine plane. B. ZELINKA [3] extends the discussion of connectivity problems in graphs also to these figurative elements. He proves a theorem of Menger type for the special class of free ends in a locally finite graph (see [1], p. 129 for the definition of a free end). I t is the purpose of this note to show t h a t in fact the restriction onto free ends is not necessary.