  The Erd\H{o}s, Gr\"unwald and Weiszfeld theorem provides a characterization
of infinite graphs which are Eulerian. That is, infinite graphs which admit
infinite Eulerian trails. In this article we complement this theorem with a
characterization of those finite trails that can be extended to infinite
Eulerian trails. This allows us to prove an effective version of the Erd\H{o}s,
Gr\"unwald and Weiszfeld theorem for a class of graphs that includes non
locally finite ones, generalizing a theorem of D.Bean.
