Recursion Relations for one-loop gravity amplitudes

We study the application of recursion relations to the calculation of finite one-loop gravity amplitudes. It is shown explicitly that the known five, and six graviton one-loop amplitudes for which the external legs have identical outgoing helicities, and the four graviton amplitude with helicities -+++ can be derived from simple recursion relations. The latter amplitude is derived by introducing a one-loop three-point vertex of gravitons of positive helicity, which is the counterpart in gravity of the one-loop three-plus vertex in Yang-Mills. We show that new issues arise for the five point amplitude with helicities -++++, where the application of known methods does not appear to work, and we discuss possible resolutions.