One of the exciting features of graphene is a possibility to affect its electrical, optical, and chemical properties by gating, that is, by application of an electric field. This requires reasonably large fields (at the level of 1 V/nm necessary to induce relevant electron density changes) applied over a gating dielectric material. At these fields, most dielectrics show some conduction, which leads to an important question: what is the best dielectric to gate graphene? Here, we show that this question is imprecise as a dielectric material produced by different fabrication methods can exhibit dramatically different gating properties. Namely, we show that two oxide dielectrics (hafnia and alumina) result in positive hysteresis of graphene gating characteristics being fabricated by atomic layer deposition and negative hysteresis being fabricated by electron beam evaporation. We attribute this behavior to the stoichiometry of the samples and oxygen ion migration. It implies that oxide dielectrics should be avoided in graphene gated devices working at room temperatures.