Distinguishing graphs with infinite motion and nonlinear growth

Cuno, Johannes; Imrich, Wilfried; Lehner, Florian

doi:10.26493/1855-3974.334.fe4

Cited by 11 publications

(23 citation statements)

References 14 publications

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“…When applied to graphs, the Motion Lemma 3.1 asserts that a graph G with motion m(G) is 2-distinguishable if m(A) ≥ 2 log 2 |A|. When m(G) is infinite, this inequality is equivalent to 2 m(G) ≥ | Aut(G)| and leads to the Motion Conjecture for Graphs of [8].…”

Section: Proposition 32 ([7]mentioning

confidence: 99%

Infinite motion and 2-distinguishability of graphs and groups

et al. 2014

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A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. When A is finite, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies that the action of Aut(X) on X is 2-distinguishable in all but finitely many instances.\ud We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee 2-distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is 2-distinguishable. One cannot extend the Motion Lemma to uncountable permutation groups, but nonetheless we prove that (under the permutation topology) every closed permutation group with infinite motion has a dense subgroup which is 2-distinguishable. We conjecture an extension of the Motion Lemma which we expect holds for a restricted class of uncountable permutation groups, and we conclude with a list of open questions. The consequences of our results are drawn for orbit equivalence of infinite permutation groups

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Section: Proposition 32 ([7]mentioning

confidence: 99%

Infinite motion and 2-distinguishability of graphs and groups

et al. 2014

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“…The aim of this paper is the presentation of fundamental results for the endomorphism distinguishing number, and of open problems. In particular, we extend the Motion Lemma of Russell and Sundaram [14] to endomorphisms, present endomorphism motion conjectures that generalize the Infinite Motion Conjecture of Tom Tucker [16] and the Motion Conjecture of [4], prove the validity of special cases, and support the conjectures by examples.…”

Section: Introductionmentioning

confidence: 69%

“…This is a generalization of the Motion Conjecture of [4] for automorphisms of graphs. Notice that we assume connectedness now, which we did not do before.…”

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confidence: 97%

Endomorphism Breaking in Graphs

Imrich

Kalinowski

Lehner

et al. 2014

Electron. J. Combin.

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We introduce the endomorphism distinguishing number D e (G) of a graph G as the least cardinal d such that G has a vertex coloring * 1 with d colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number D(G) of a graph G, which is defined for automorphisms instead of endomorphisms.As the number of endomorphisms can vastly exceed the number of automorphisms, the new concept opens challenging problems, several of which are presented here. In particular, we investigate relationships between D e (G) and the endomorphism motion of a graph G, that is, the least possible number of vertices moved by a nontrivial endomorphism of G. Moreover, we extend numerous results about the distinguishing number of finite and infinite graphs to the endomorphism distinguishing number. This is the main concern of the paper.

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“…Note that if the motion of such a graph is infinite, then then it must be ℵ 0 and that 2 ℵ 0 is a trivial upper bound for the size of the automorphism group. While Tucker's conjecture is still wide open, there are many partial results towards it, see [3,6,7,10,11,14,16]. Broere and Pilśniak [2] noticed that most of these partial results can be generalised to edge colourings.…”

Section: Introductionmentioning

confidence: 99%