A new proof of validity of Bouchet's conjecture on Eulerian bidirected graphs

Recently, E. M'{a}v{c}ajov'{a} and M. v{S}koviera proved that every bidirected Eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. This result shows the validity of Bouchet's nowhere zero conjecture for Eulerian bidirected graphs. In this paper we prove the same theorem in a different terminology and with a short and simple proof. More precisely, we prove that every Eulerian undirected graph which admits a zero-sum flow, admits a zero-sum $4$-flow. As a conclusion we obtain a shorter proof for the previously mentioned result of M'{a}v{c}ajov'{a} and v{S}koviera.