Melanin is the pigment found in human skin that is responsible for both photoprotection and photodamage. Recently there have been reports that greater photodamage of DNA occurs when cells containing melanin are irradiated with ultraviolet (UV) radiation, thus suggesting that the photoproducts of melanin cause DNA damage. Photoionization processes have also been implicated in the photodegradation of melanin. However, not much is known about the oxidation potential of melanin and its monomers. In this work we calculate the ionization energies of monomers, dimers, and few oligomers of eumelanin to estimate the threshold energy required for the ionization of eumelanin. We find that this threshold is within the UV-B region for eumelanin. We also look at the charge and spin distributions of the various ionized states of the monomers that are formed to understand which of the ionization channels might favor monomerization from a covalent dimer.