We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of an external magnetic field H. We compute the number of singlespin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H − m plane (focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between −∞ and +∞ at T = 0. The annealed complexity ΣA(m, H) is calculated for z = 2, 3, 4 and the quenched complexity ΣQ(m, H) for z = 2. We prove explicitly for z = 2 that the contour ΣQ(m, H) = 0 coincides with the saturation loop. On the other hand, we show that ΣA(m, H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.