Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What about special cases?
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In general, you can't say mauch. Take $\Gamma$ to be the complete graph $K_n$. Then $\Gamma$ is a Cayley graph for any group $K$ of order $n$, and any group $H$ of order $n/2$ acts semiregularly on $\Gamma$ with two orbits. |
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EDIT: I assumed that $H$ was a subgroup of $K$, which was not in the question. No, there are no other relations: given a group $K$, a subgroup $H$ of index $2$ and a Cayley graph $\Gamma$ on $K$, it is always the case that $H$ is a semiregular subgroup of $Aut(\Gamma)$ with two orbits. In fact, there is nothing special about $2$ here. If $\Gamma$ is a Cayley graph on $K$, then $K$ acts regularly on the vertices of $\Gamma$. In particular, any subgroup $H$ of $K$ will act semiregularly on the vertices, and the orbits of $H$ will simply be the cosets of $H$ in $K$. In particular, the number of orbits is the index of $H$ in $K$. This all follows directly from the definitions and, as such, not really research level. |
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