For $n\in\mathbb{N}$ let $S_n$ denote the set of permutations on the set $\{1,\ldots,n\}$. Set $$E_n = \big\{\{\pi_1, \pi_2\}: \pi_1,\pi_2\in S_n \land \exists k_1 < k_2 <\ldots <k_r\leq n: \pi_2=(k_1 \cdots k_r)\circ \pi_1\big\}.$$
(In other words, $\pi_2$ can be generated from $\pi_1$ with a **monotonic** (or monotonic like) cyclic permutation.)

Let $G_n=(S_n, E_n)$. Given $n\in\mathbb{N}$, what is $\chi(G_n)$?