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Distinguishing threshold of graphs

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 a coloring, and the distinguishing threshold of $G$, denoted by $θ(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. As an alternative definition, $θ(G)$ is one more than the maximum number of cycles in the cycle decomposition of automorphisms of $G$. In this paper, we characterize $θ(G)$ when $G$ is disconnected. Afterwards, we prove that, although for every positive integer $k\neq 2$ there are infinitely many graphs whose distinguishing thresholds are equal to $k$, we have $θ(G)=2$ if and only if $\vert V(G)\vert =2$. Moreover, we show that if $θ(G)=3$, then either $G$ is isomorphic to one of the four graphs on~3 vertices or it is of order $2p$, where $p\neq 3,5$ is a prime number. Furthermore, we prove that $θ(G)=D(G)$ if and only if $G$ is asymmetric, $K_n$ or $\overline{K_n}$. Finally, we consider all generalized Johnson graphs, $J(n,k,i)$, which are the graphs on all $k$-subsets of $\{1,\ldots , n\}$ where two vertices $A$ and $B$ are adjacent if $|A\cap B|=k-i$. After studying their automorphism groups and distinguishing numbers, we calculate their distinguishing thresholds as $θ(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$, unless $ k=\frac{n}{2}$ and $i\in\{ \frac{k}{2} , k\}$ in which case we have $θ(J(n,k,i))={n\choose k}$.