The effects of cooling conditions on graphene growth by the atmospheric pressure chemical vapor deposition (APCVD) method on platinum are investigated. It is found that the cooling conditions are the key to better control of the coverage, thickness and uniformity of the graphene films. A series of experiments were carried out with different cooling rates and gas atmospheres. Various characterization techniques (SEM, Raman, AFM and TEM) were applied to examine the graphene morphology and quality. A "three competitive reactions" mechanism is proposed to explain the cooling condition effects. Based on optimized growth conditions, a high-coverage, single-layer graphene film could be grown on Pt using the APCVD method. Back-gated field effect transistors were fabricated and the measured carrier mobility is about 1600 cm(2) V(-1) s(-1).