Distinguishing threshold of graphs

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms can preserve it. The number $G$, denoted by $D(G)$, the minimum colors required for such coloring, and threshold $\theta(G)$, $k$ that every $k$-coloring distinguishing. As an alternative definition, $\theta(G)$ one more than maximum cycles in cycle decomposition $G$. In this paper, we characterize $\theta (G)$ when disconnected. Afterwards, prove that, although positive integer $k\neq 2$ there are infinitely many graphs whose thresholds equal to $k$, have $\theta(G)=2$ only $\vert V(G)\vert =2$. Moreover, show $\theta(G)=3$, then either isomorphic four on~3 vertices or it order $2p$, where $p\neq 3,5$ prime number. Furthermore, $\theta(G)=D(G)$ asymmetric, $K_n$ $\overline{K_n}$. Finally, consider all generalized Johnson graphs, $J(n,k,i)$, which on $k$-subsets $\{1,\ldots , n\}$ two $A$ $B$ adjacent $|A\cap B|=k-i$. After studying their automorphism groups numbers, calculate as $\theta(J(n,k,i))={n\choose k} - {n-2\choose k-1}+1$, unless $ k=\frac{n}{2}$ $i\in\{ \frac{k}{2} k\}$ case k}$.