Difference in chromatic number between Schreier coset graphs and Cayley graphs Can the Schreier coset 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.
 A: 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 $n$.
A: As requested by the OP, here is a simple example of the fact that the chromatic number may go up, and that the Schreier graph is not a subgraph of the Cayley graph.
Let $k>2$ be odd and $n>1$ be even. Let $G = C_{kn}$ (the cyclic groups on $kn$  elements, it is $\cong \mathbb{Z}/kn\mathbb{Z}$).
Let $S = \lbrace -1, 1\rbrace$ be a [symmetric] generating set of this cyclic group.
Then $\mathrm{Cay}(G,S)$ is a cycle of length $kn$ (in particular it contains no odd cycles, since $n>1$).
There is a [normal] subgroup $H$ in $G$ which is isomorphic to $C_n$ (this is the subgroup generated by $k \in \mathbb{Z}$).
It is fairly standard exercise to check that $G/H \cong C_k$.
By normality $\mathrm{Sch}(G/H,S)$ is isomorphic $\mathrm{Cay}(G/H,S)$ which itself is a cycle of length $k$.
The Schreier Graph $\mathrm{Sch}(G/H,S)$, being an odd cycle, has chromatic number 3 and is not isomorphic to any subgraph of $\mathrm{Cay}(G,S)$ (which is an even cycle with chromatic number 2).
Furthermore, by fixing $k$ an letting $n \to \infty$, the ratio $\frac{\# \text{Vertices of } \mathrm{Cay}(G,S)}{\#\text{Vertices of } \mathrm{Sch}(G/H,S)}$ tends to infinity. By fixing $n$ and letting $k \to \infty$, the ratio remains constant, while the cardinalities tend to infinity.
