A bipartite graph's toric edge ideal satisfies Green-Lazarsfeld N_p exactly when the graph's missing-edge complement is a small tree for p=2, the graph is complete bipartite for p=3, or the graph is K_{2,n} for p at least 4.
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Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs
A bipartite graph's toric edge ideal satisfies Green-Lazarsfeld N_p exactly when the graph's missing-edge complement is a small tree for p=2, the graph is complete bipartite for p=3, or the graph is K_{2,n} for p at least 4.