The concept of fuzzy chromatic number (FCN) is being applied, which is of fuzzy graphs that is constructed by fuzzy independent vertex sets. In this, any two fuzzy graphs, G1 and G2, were assumed, where the vertex sets of these two fuzzy graphs (G1 and G2) are disjoint. In this study, initially, we found the fuzzy chromatic number (FCN) of cartesian and tensor product, then we developed the result of fuzzy chromatic number (FCN) of cartesian and tensor product. After that, to find the fuzzy chromatic number (FCN) of the cartesian and tensor product, an algorithm is being developed according to the result obtained.