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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.

We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree and Castelnuovo--Mumford regularity. The main tool is a Gr\"obner basis of the ideal obtained combinatorially from the hypergraph. We define the notion of parity join in a hypergraph and show that the regularity of the Eulerian ideal is equal to the maximum cardinality of such a set of edges. The formula for the degree involves the cardinality of the set of sets of vertices, $T$, that admit a $T$-join. We compute the degree and regularity explicity in the cases of a complete $k$-partite hypergraph and a complete hypergraph of rank $3$.