Chemical vapor deposition has emerged as a promising approach for high-quality graphene synthesis. Graphene quality, including the defect density, the size of monocrystal domains, and the number of layers, depend on the growth conditions. The main experimental parameters include temperature, metal catalyst composition, growth atmosphere, and gaseous dynamics in the reaction chamber. Here we show, that even for the same growth conditions, simply changing carbon source from methane to ethane or acetylene yields drastically different growth mode that allows the production of single-crystal-like graphene on a polycrystalline metal foil substrate. This new single crystal growth mechanism does not require local supply of carbon precursors or special single crystal catalytic substrate that were introduced earlier. We assign the observed differences in the growth mechanism to formation of stable and mobile dimeric C2Hx intermediate species from ethane and acetylene, in contrast to less mobile CHx and C3Hx species formed from methane and propane.