An upper bound over the number of bipolar orientations for a regular graph Given a $k$-regular graph $G$, the number of acyclic orientations $Acy(G)$ is $\chi(-1)$ where $\chi$ is the chromatic polynomial of $G$. How many bipolar orientations does $G$ have? 

Is there an upper bound for it? I assume it should be exponentially lower than $Acy(G)$ but didn't succeed in finding a known result connecting these two numbers.
It is not always lower than the number of acyclic orientations by any factor, let alone an exponential one. In particular, in a complete graph, there are $n!$ acyclic orientations, all of which are bipolar.