In-plane heterostructure of hexagonal boron nitride and graphene (h-BN-G) has become a research focus of graphene due to its predicted fascinating properties such as bandgap opening and magnetism, which hence has ignited the attempt of experimentally growing such in-plane two-dimensional (2D) hybrid materials. Many previous researches demonstrated the synthesis of such heterostructures on Cu foils via chemical vapor deposition (CVD) process. The obtained 2D hybrid materials would offer a possibility for fabricating atomically thin electronic devices. However, many fundamental issues are still unclear, including the in-plane atomic continuity, the edge type, and the electronic properties at the boundary of hybridized h-BN and graphene domain. To clarify these issues, we report the syntheses of h-BN-G monolayer heterostructures on strongly coupled Rh(111) substrate and weakly coupled Ir(111) substrate via a two-step growth process in an ultrahigh vacuum (UHV) system, respectively. With the aid of scanning tunneling microscopy (STM), it is revealed that graphene and h-BN could be linked together seamlessly on an atomic scale at the linking boundaries. More importantly, we find that the atomically sharp zigzag-type boundaries dominate the patching interface between graphene and h-BN as demonstrated by atomic-scale STM images. To understand the physical origin of the atomic linking of the h-BN-G heterostructures, we also perform density functional theory (DFT) calculations, including geometry optimizations and binding energy calculations for different kinds of linking interfaces. The calculated results reconfirm that graphene prefers to grow on the h-BN domain edges and form zigzag linking boundaries. Besides the atomic structures on the linking interfaces, the electronic characteristics are also of particular importance. It is worth noting that the substrates coupled strongly with graphene by π-d orbital hybridization (such as Rh(111) and Ru(0001)), lead to downward shift of graphene π-bands away from the Fermi level, or decay of the intrinsic electronic structure of graphene. In this regard, the influence of h-BN on the electronic property of graphene is hard to identify on such h-BN-G heterostructures. The weakly coupled Ir(111) is chosen to be a perfect substrate to investigate the interface electronic properties of h-BN-G heterostructure due to the absence of substrate electronic doping effect. Scanning tunneling spectroscopy studies indicate that the graphene and h-BN tend to exhibit their own intrinsic electronic features near the linking boundaries on Ir(111). Therefore, the present work offers a deep insight into the h-BN-G boundary structures and the effect of adlayer-substrate coupling both geometrically and electronically.