Distinguishability of Locally Finite Trees

The distinguishing number ∆(X) of a graph X is the least positive integer n for which there exists a function f : V (X) → {0, 1, 2, · · · , n−1} such that no nonidentity element of Aut(X) fixes (setwise) every inverse image f−1(k), k ∈ {0, 1, 2, · · · , n − 1}. All infinite, locally finite trees without pendant vertices are shown to be 2distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree T with finite distinguishing number contains a finite subtree J such that ∆(J) = ∆(T ). Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer n such that the function f is also a proper vertex-coloring.