Distinguishing graphs with intermediate growth

Abstract

A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finite graph with infinite motion and growth at most O 2 (1−ε) √ n 2 is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.

Random Colourings and Automorphism Breaking in Locally Finite Graphs

“…The proof stated in Section 5 also works for this result. The second auxiliary lemma we will use can be found in [10], although the implications stated below have been known before; see for example [2], where they play a crucial role in the proof of one of the main results. Lemma 6.18.…”

Trees with distinguishing index equal distinguishing number plus one

“…The introduction of the distinguishing number in 1996 by Albertson and Collins [1] was a great success; by now about one hundred papers were written motivated by this seminal paper! The core of the research has been done on the invariant D itself, either on finite [6,11,14] or infinite graphs [9,16,18]; see also the references therein. Extensions to group theory (cf.…”

Distinguishing Graphs of Maximum Valence 3

“…The second equally important motivation is the Infinite Motion Conjecture of Tucker [28], who conjectured that each connected, locally finite infinite graph is 2-distinguishable, if every automorphism that is not the identity moves infinitely many vertices. The conjecture is still open, although it has been shown to be true for many classes of graphs [9,17,27], in particular for graphs of subexponential growth [22], and thus for all graphs of polynomial growth. For a long time it was not clear whether it holds for graphs of maximal valence 3, and whether infinite motion was really needed.…”