Multi-Faithful Spanning Trees of Infinite Graphs

Abstract

For an end τ and a tree T of a graph G we denote respectively by m(τ) and m _T (τ) the maximum numbers of pairwise disjoint rays of G and T belonging to τ, and we define tm(τ) := min{m _T(τ): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end τ of G to a cardinal f(τ) such that tm(τ) ⩽ f(τ) ⩽ m(τ), there exists a spanning tree T of G such that m _T (τ) = f(τ) for every end τ of G.