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?



Semiregular subgroups of automorphism group of cayley graphs
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.


