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Advances in Fuzzy Systems
Volume 2019, Article ID 5213020, 11 pages
Research Article

Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using α-Cuts

Department of Mathematics, Wollega University, Nekemte, Ethiopia

Correspondence should be addressed to Mamo Abebe Ashebo; moc.liamg@73ebebaomam

Received 24 January 2019; Revised 28 March 2019; Accepted 2 April 2019; Published 2 May 2019

Academic Editor: Antonin Dvorák

Copyright © 2019 Mamo Abebe Ashebo and V. N. Srinivasa Rao Repalle. This 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 traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on -cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first 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.