  Given a graph $G$ with only even degrees let $\varepsilon(G)$ denote the
number of Eulerian orientations, and let $h(G)$ denote the number of half
graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$.
Recently, Borb\'enyi and Csikv\'ari proved that $\varepsilon(G)\geq h(G)$ holds
true for all Eulerian graphs with equality if and and only if $G$ is bipartite.
In this paper we give a simple new proof of this fact, and we give identities
and inequalities for the number of Eulerian orientations and half graphs of a
$2$-cover of a graph $G$.
