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A vertex coloring of a graph G is called distinguishing if no non-identity automorphisms of G can preserve it. The distinguishing number of G, denoted by D(G), is the minimum number of colors required for such coloring. The distinguishing threshold of G, denoted by θ(G), is the minimum number k of colors such that every k-coloring of G is distinguishing. In this paper, we study θ(G), find its relation to the cycle structure of the automorphism group and prove that θ(G) = 2 if and only if G is isomorphic to K 2 or K 2 . Moreover, we study graphs that have the distinguishing threshold equal to 3 or more and prove that θ(G) = D(G) if and only if G is asymmetric, K n or K n . Finally, we consider Johnson scheme graphs for their distinguishing number and threshold concludes the paper.

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