  Associated to any hypergraph is a toric ideal encoding the algebraic
relations among its edges. We study these ideals and the combinatorics of their
minimal generators, and derive general degree bounds for both uniform and
non-uniform hypergraphs in terms of balanced hypergraph bicolorings,
separators, and splitting sets. In turn, this provides complexity bounds for
algebraic statistical models associated to hypergraphs. As two main
applications, we recover a well-known complexity result for Markov bases of
arbitrary 3-way tables, and we show that the defining ideal of the tangential
variety is generated by quadratics and cubics in cumulant coordinates.
