Distinguishing labeling of the actions of almost simple groups

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 xX 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