Covers, orientations and factors (Q785575)

Summary: 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, \textit{M. Borbényi} and \textit{P. Csikvári} [Discrete Math. 343, No. 6, Article ID 111842, 14 p. (2020; Zbl 1437.05101)] proved that \(\varepsilon(G)\geqslant h(G)\) holds true for all Eulerian graphs, with equality if 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 \(G\).