Smith, Simon M. and Watkins, Mark E. (2014) Bounding the distinguishing number of infinite graphs and permutation groups. Electronic Journal of Combinatorics, 21 (3). p3.40. ISSN 10778926
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Abstract
A group of permutations G of a set V is kdistinguishable if there exists a partition of V into k cells such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G, V) is kdistinguishable is its distinguishing number, D(G, V). In particular, a graph X is kdistinguishable if its automorphism group Aut(X) satisfies D(Aut(X), VX) ≤ k.
Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show first that whenever an infinite connected graph X is not kdistinguishable (for a given cardinal k), then it contains a ball of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k ≥ 3 there exists an infinite, locally finite, connected graph X that is not kdistinguishable but in which every ball of finite radius is kdistinguishable.
In the second half of this paper we show that a large distinguishing number for an imprimitive permutation group G is traceable to a high distinguishing number either of a block of imprimitivity or of the action induced by G on the corresponding system of imprimitiv ity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.
Keywords:  distinguishing number, distinguishing coloring, JCNotOpen 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
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ID Code:  25001 
Deposited On:  16 Nov 2016 20:55 
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