Asymmetrizing trees of maximum valence $$2^{\aleph _0}$$

Abstract

Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity automorphisms of T. We give bounds on the maximum valence d of T that assure the existence of a vertex coloring of T with two colors that is preserved only by the identity automorphism. For finite m we obtain the bound $$d\le 2^{m/2}$$d≤2m/2 when T is finite, and $$d\le 2^{(m-2)/2}+2$$d≤2(m-2)/2+2 when T is infinite. For countably infinite m the bound is $$d\le 2^m.$$d≤2m. This relates to a question of Babai, who asked whether there existed a function f(d) such that every connected, locally finite graph G with maximum valence d has a 2-coloring of its vertices that is only preserved by the identity automorphism if the minimum number m of vertices moved by each non-identity automorphisms of G is at least $$m\ge f(d)$$m≥f(d). Our results give a positive answer for trees. The trees need not be locally finite, their maximal valence can be $$2^{\aleph _0}$$2ℵ0. For finite m we also extend our results to asymmetrizing trees by more than two colors.

On Asymmetric Colourings of Claw-Free Graphs

“…For trees f (d) = 2 log 2 d , see [9]. Let us also mention that Lehner and Verret [16] recently determined all finite, 4-regular, connected, vertex-transitive graphs G with D(G) > 2.…”