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 ۴-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 ۴-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.

Keywords:

Nowhere zero flow in bidirected graphs , zero-sum flow , Eulerian graphs