Asymmetrizing trees of maximum valence 2ℵ0

Wilfried Imrich*, Thomas W. Tucker

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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≤ 2 m / 2 when T is finite, and d≤ 2 ( m - 2 ) / 2+ 2 when T is infinite. For countably infinite m the bound is d≤ 2 m. 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≥ f(d). Our results give a positive answer for trees. The trees need not be locally finite, their maximal valence can be 2ℵ0. For finite m we also extend our results to asymmetrizing trees by more than two colors.

Original languageEnglish
Pages (from-to)615-624
Number of pages10
JournalMonatshefte für Mathematik
Volume192
Issue number3
DOIs
Publication statusPublished - 28 Jul 2020
Externally publishedYes

Keywords

  • Asymmetric colorings
  • Automorphisms
  • Finite and infinite trees

ASJC Scopus subject areas

  • Mathematics(all)

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