## Abstract

Suppose *Γ* is a group acting on a set *X*, written as (*Γ,X*). An *r*-labeling *f: X*→{1,2, ..., *r*} of *X* is called distinguishing for (*Γ,X*) if for all *σ∈Γ,σ*≠1, there exists an element *x*∈*X* such that *f*(*x*)≠*f*(*x*
^{σ}). The distinguishing number *d*(*Γ,X*) of (*Γ,X*) is the minimum *r* for which there is a distinguishing *r*-labeling for (*Γ,X*). If *Γ* is the automorphism group of a graph *G*, then *d*(*Γ,V* (*G*)) is denoted by *d*(*G*), and is called the distinguishing number of the graph *G.* The distinguishing set of *Γ*-actions is defined to be *D**(*Γ*)={*d*(*Γ,X*): *Γ* acts on *X*}, and the distinguishing set of *Γ*-graphs is defined to be *D*(*Γ*)={*d*(*G*): Aut(*G*)≅*Γ*}. This paper determines the distinguishing set of *Γ*-actions and the distinguishing set of *Γ*-graphs for almost simple groups *Γ.*

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Supported in part by the NSF and by ARC Grant DP1096525

Supported in part by the National Science Council under grant NSC92-2115-M-110-010

Supported in part by ZJNSF under grant Z6110786.

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Seress, Á., Wong, TL. & Zhu, X. Distinguishing labeling of the actions of almost simple groups.
*Combinatorica* **31, **489–506 (2011). https://doi.org/10.1007/s00493-011-2221-7

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### Mathematics Subject Classification (2000)

- 20G15
- 05C25
- 20B25