The mixed spin-2 and spin-5/2 Blume-Emery-Griffiths (BEG) Ising ferrimagnetic system is studied on the Bethe lattice using the exact recursion equations with the coordination number q = 3 corresponding to the honeycomb lattice on real lattices. The influences of the crystal field and the biquadratic interactions are investigated by obtaining the phase diagrams on the (K/vertical bar J vertical bar,kT/vertical bar J vertical bar) and (D1/vertical bar J vertical bar kT/vertical bar J vertical bar) planes, respectively, with equal crystal field interactions for the sublattices. The model presents very rich critical behaviors, which includes the first- and second-order phase transitions, thus also the tricritical and critical end points are observed. We have also found that the model gives up to four compensation temperatures for appropriate values of the system parameters. (c) 2006 Published by Elsevier B.V.