We present our first-principles calculation of the adsorption and diffusion of a carbon adatom on the H-terminated and clean Ge(110) surfaces, which are essential processes in the nucleation and growth of a monolayer graphene on Ge(110) by chemical vapor deposition. On the H-terminated surface, the C adatom spontaneously substitutes H atom(s) to form a monohydride structure (CH) or a dihydride structure (CH2) and makes direct bonds with the substrate Ge atoms. The resulting diffusion barriers of the C adatom are 2.67 and 6.45 eV parallel to and perpendicular to the zigzag Ge chains of the surface, respectively. On the clean surface, the C adatom embeds into the zigzag Ge chain with nearly no barrier, kicking out a Ge atom out of the chain at the same time. The kicked-out Ge atom, instead of the C adatom, becomes a diffusion species with the barrier less than 0.63 eV. The formation of the C composite structures makes the C adatom difficult to diffuse both on the H-terminated and clean Ge(110) surfaces, which suggests that the nucleation and growth of the graphene islands from C seeds is much suppressed. We propose a growth mechanism of graphene monolayer going round the diffusion of the C adatoms on the Ge(110) surfaces. (C) 2017 Elsevier B.V. All rights reserved.