We use the notion of fuzzy chromatic number (FCN) of fuzzy graphs based on fuzzy independent vertex sets introduced in 2015. Let be a path fuzzy graph and be any fuzzy graphs where their vertex sets are disjoint. Let be a cartesian product of and . In this paper, we construct formula for FCN of and verify connection between maximum of FCN of both fuzzy graphs and FCN of their cartesian product. Also, we create an algorithm to determine FCN of the cartesian product according to the properties obtained. The last two statements show novelties of the present work. Evaluation of the algorithm is presented in the experimental results.
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