Covers, orientations and factors

Given a graph $G$ with only even degrees let $\epsilon (G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)\mathrm{/}2$ for each vertex $v$. Recently, Borb'enyi and Csikv'ari proved that $\epsilon (G)\ge h(G)$ 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 $G$.