The decomposition of methane is an environmentally attractive approach to CO and CO₂-free hydrogen production. In this work, we use Density Functional Theory and Statistical Thermodynamics to investigate the growth of graphene in a methane atmosphere in the absence of metallic particles. Our results indicate that the graphene edge itself can be used as a catalyst for methane decomposition, as the active sites are continually regenerated by the deposition of carbon from methane. Furthermore, since no other gases are produced in the process, the costs of CO₂ sequestration and hydrogen purification from CO are eliminated. We also discuss the conditions at which this process is thermodynamically favorable.