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.
A vertex coloring of a graph G is called distinguishing if no nonidentity automorphisms of G can preserve it. The distinguishing number of G, denoted by D G ( ), is the
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.