  We study the minimal homogeneous generating sets of the Eulerian ideal
associated with a simple graph and its maximal generating degree. We show that
the Eulerian ideal is a lattice ideal and use this to give a characterization
of binomials belonging to a minimal homogeneous generating set. In this way, we
obtain an explicit minimal homogeneous generating set. We find an upper bound
for the maximal generating degree in terms of the graph. This invariant is half
the number of edges of a largest Eulerian subgraph of even cardinality without
even-chords. We show that for bipartite graphs this invariant is the maximal
generating degree. In particular, we prove that if the graph is bipartite, the
Eulerian ideal is generated in degree $2$ if and only if the graph is chordal.
Furthermore, we show that the maximal generating degree is also $2$ when the
graph is a complete graph.
