Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam\nscheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and\ndefined based on alpha -cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type\nwas fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge\nset. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy\nchromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy\nchromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.
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