If you assume as the "melting" of graphene the decomposition of the 2D crystal in a 3D network of 1D chains, then calculations have shown that is Tm ≈ 4900 K.
Antibacterial properties of semiconductor material (eg; TiO2, ZnO etc) can be attributable to reaction of holes or electrons with OH-, O2 in water to form radicals. These radicals will be the elements responsible to eliminate the bacteria. Since the semiconductor has band gap, energy (usually from UV) is required to excite the electron to jump from valence to conduction band, leaving holes in the valence band. Due to instability, the excited electrons will tend to fall back and recombine with the holes in the valence band, releasing energy in the form of heat. In this case, since there is no free electron or hole to react with O2 or OH-, radicals will not form. Hence the antibacterial properties will not be established. So the main problem here is recombination of electron to the hole. We do not want this to happen because radicals wont form. To overcome this problem, we couple or dope it with metals( typically noble metals even ordinary metals is OK). The best choice is to have metal where its Fermi level is lower than the conduction band of the semiconductor. By doing this, the electron excited in the conduction band can be moved into the metals when it looses energy instead of going down to valence band and recombine with hole. I have no experience with graphene, but I think its Fermi level is OK.