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.

My question is, 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.

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.

The Schreier coset graphs can be seen as a subgraph of Cayley graph on the same groups and, hence, have their chromatic numbers bounded by the chromatic numbers of the Cayley graphs on those groups with the same generating set.

My question is, 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.

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.

My question is, 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.