I will review the connection between on-shell diagram formulation of N=4 scattering amplitudes, the Grassmannian, and N=1 bipartite field theories. I will then perform a detailed investigation of the combinatorial and geometric objects associated to bipartite graphs, focusing on toric geometry and on the stratification of the Grassmannian. I will show how this description extends to the non-planar case, providing powerful tools to study the on-shell formulation of scattering amplitudes beyond the planar limit.