Can the [Schreier coset](https://en.wikipedia.org/wiki/Schreier_coset_graph) graphs can be seen as a subgraph of Cayley graph on the same groups(neglecting the loop edges) and, hence, have their chromatic numbers bounded by the chromatic numbers of the Cayley graphs on those groups with the same generating set?


Also, how much does the cardinality of the subgroup determine the gap between the chromatic number of schreier coset graphs and the Cayley graphs on those same groups with the same generating set. Like, if the subgroup with respect to which the cosets are taken is large, then is the gap between the chromatic numbers also proportionally large? Thanks beforehand.



Difference in chromatic number between Schreier coset graphs and Cayley graphs
Here's a good example.


Let $G=S\_n$, $S=\{(i,j)\mid 1\leq i<j\leq n \}$. Then $\operatorname{Cay}(G,S)$ is a bipartite graph and so its chromatic number is $2$. Let $H=\{g\in S\_n\mid g(n)=n\}$. It is easy to check that $\operatorname{Sch}(G/H,S)$ is a complete graph (if we forget about loops) and hence its chromatic number is $n$.


