  We investigate the class of bipartite Borel graphs organized by the order of
Borel homomorphism. We show that this class is unbounded by finding a jump
operator for Borel graphs analogous to a jump operator of Louveau for Borel
equivalence relations. The proof relies on a non-separation result for iterated
Frechet ideals and filters due to Debs and Saint Raymond. We give a new proof
of this fact using effective descriptive set theory. We also investigate an
analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not
yield a jump operator for bipartite Borel graphs. However, we use it to answer
a question of Kechris and Marks by showing that there is a Borel graph with no
Borel homomorphism to a locally countable Borel graph, but each of whose
connected components has a countable Borel coloring.
