Schreier coset graph

Remarks

The rule is well defined: if $Hg = Hg'$, then $g' = hg$ for some $h \in H$, so $Hg's = Hhgs = Hgs$. The number of vertices is the index of $H$ in $G$, and every vertex has exactly one outgoing edge with each label. If $S$ generates $G$, the graph is connected. For $H = \set{1}$ it is the Cayley graph of $G$ with respect to $S$.

Example: let $G = \Z$, $H = 3\Z$ and $S = \set{1}$. The vertices are $3\Z$, $1 + 3\Z$ and $2 + 3\Z$, and the edges go $3\Z \to 1 + 3\Z \to 2 + 3\Z \to 3\Z$, since adding $1$ moves each coset to the next one. The graph is a directed cycle of length $3$.

Non-example: the rule with left cosets $gH$ and edges $gH \to gsH$ is not well defined in general. In $S_3$, with composition from right to left, take $H = \set{1, (1\,2)}$ and $s = (2\,3)$. Then $1 \cdot H = (1\,2) H$, but $(2\,3) H = \set{(2\,3), (1\,3\,2)}$ and $(1\,2)(2\,3) H = (1\,2\,3) H = \set{(1\,2\,3), (1\,3)}$ are different cosets.