Research Article Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using -Cuts Mamo Abebe Ashebo and V. N. Srinivasa Rao Repalle Department of Mathematics, Wollega University, Nekemte, Ethiopia Correspondence should be addressed to Mamo Abebe Ashebo; mamoabebe37@gmail.com Received 24 January 2019; Revised 28 March 2019; Accepted 2 April 2019; Published 2 May 2019 Academic Editor: Antonin Dvor´ ak Copyright © 2019 Mamo Abebe Ashebo and V. N. Srinivasa Rao Repalle. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like trafc light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defned based on -cuts of fuzzy graph. Two diferent types of fuzziness to fuzzy graph are considered in the paper. Te frst type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained. 1. Introduction Nowadays, many real world problems cannot be properly modeled by a crisp graph theory, since the problems contain uncertain information. Te fuzzy set theory, anticipated by Zadeh [1], is used to handle the phenomena of uncertainty in real life situation. A lot of works have been done in fuzzy shortest path problems using type 1 fuzzy set in [2–5]. Dey et al. [6] introduced interval type 2 fuzzy set in the fuzzy shortest path problems. Recently, in [7], the authors proposed a genetic algorithm for solving fuzzy shortest path problem with interval type 2 fuzzy arc lengths. Some researchers also used the fuzzy set theory to touch the uncertainty in crisp graphs. Kaufmann [8] proposed the frst defnition of fuzzy graph in 1973, based on Zadeh’s fuzzy relations. Later, Rosenfeld [9] introduced another elaborated defnition of fuzzy graph with fuzzy vertex set and fuzzy edge set in 1975. He developed the theory of fuzzy graph. Afer that, Bhattacharya [10] has established some connectivity concepts regarding fuzzy cut nodes and fuzzy bridges. Bhutani [11] has studied automorphisms on fuzzy graphs and certain properties of complete fuzzy graphs. Also, Mordeson and Nair [12] introduced cycles and cocycles of fuzzy graphs. Several authors including Sunitha and Vijayakumar [13, 14], Bhutani and Rosenfeld [15], Mathew and Sunitha [16], Akram [17], and Akram and Dudek [18] have introduced numerous concepts in fuzzy graphs. Fuzzy graph theory has several applications in various felds like clustering analysis, database theory, network analysis, information theory, etc. [19]. Coloring of fuzzy graphs plays a vital role in theory and practical applications. It is mainly studied in combina- torial optimization problems like trafc light control, exam scheduling, register allocation, etc. [20]. Fuzzy coloring of a fuzzy graph was defned by authors Eslahchi and Onagh in 2004 and later developed by them as fuzzy vertex coloring [21] in 2006. Another approach of coloring of fuzzy graphs was introduced by Munoz et al. [22]. Te authors have defned the chromatic number of a fuzzy graph. Incorporating the above two approaches of coloring of fuzzy graph, Kishore and Sunitha [23] introduced chromatic number of fuzzy graphs and developed algorithm. Dey and Pal [24] intro- duced the vertex coloring of a fuzzy graph using -cuts. In [25], they have used the vertex coloring of fuzzy graph to classify the accidental zone of a trafc control. Further, in [26], the authors proposed genetic algorithm to fnd the robust solutions for fuzzy robust coloring problem. Te authors, Ananthanarayanan and Lavanya [20], introduced fuzzy chromatic number and fuzzy total chromatic number Hindawi Advances in Fuzzy Systems Volume 2019, Article ID 5213020, 11 pages https://doi.org/10.1155/2019/5213020