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 $\chi(G\_n)$?



Chromatic number of graph defined on the set of permutations
In case it helps here is a Sage function that should generate these graphs (if I got the definition right)



```
def g(n):
    gens = []
    A = SymmetricGroup(n)
    for el in A:
        c = el.cycles()
        if len(c) == 1 and sorted(c[0]) == list(c[0]):
            gens += [el]           
    return Graph(A.cayley_graph(generators=gens, simple=True))

```

The chromatic number of $G\_2,G\_3,G\_4$ and $G\_5$ respectively seems to be $2,6,6,30.$


