Distinguishing density and the Distinct Spheres Condition

Wilfried Imrich, Florian Lehner, Simon M. Smith

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)


If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that G has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite. The Infinite Motion Conjecture is a well-known open conjecture about 2-distinguishability. A graph G is said to have infinite motion if every nontrivial automorphism of G moves infinitely many vertices, and the conjecture states that every connected, locally finite graph with infinite motion is 2-distinguishable. In this paper we show that for many classes of graphs for which the Infinite Motion Conjecture is known to hold, the graphs have 2-distinguishing density zero.

Original languageEnglish
Article number103139
JournalEuropean journal of combinatorics
Issue numberOctober
Early online date30 Apr 2020
Publication statusPublished - Oct 2020

Cite this