Advances in Fuzzy Systems

Volume 2019, Article ID 5213020, 11 pages

https://doi.org/10.1155/2019/5213020

## 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.

#### Abstract

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.

#### 1. Introduction

Nowadays, many real world problems cannot be properly modeled by a crisp graph theory, since the problems contain uncertain information. The 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 first definition of fuzzy graph in 1973, based on Zadeh’s fuzzy relations. Later, Rosenfeld [9] introduced another elaborated definition of fuzzy graph with fuzzy vertex set and fuzzy edge set in 1975. He developed the theory of fuzzy graph. After 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 fields 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 combinatorial optimization problems like traffic light control, exam scheduling, register allocation, etc. [20]. Fuzzy coloring of a fuzzy graph was defined 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]. The authors have defined 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] introduced 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 traffic control. Further, in [26], the authors proposed genetic algorithm to find the robust solutions for fuzzy robust coloring problem. The authors, Ananthanarayanan and Lavanya [20], introduced fuzzy chromatic number and fuzzy total chromatic number of a fuzzy graph using -cuts. Rosyida et al. [27] proposed a new approach to determine fuzzy chromatic number of fuzzy graph through its -chromatic number. Samanta et al. [28] also introduced a new concept for coloring of fuzzy graphs using fuzzy color. Lately, Dey et al. [29] introduced the concept of vertex and edge coloring of vague graphs which are the generalized structure of fuzzy graphs.

In the literature, to the best of our knowledge, there is no study on the fuzzy chromatic polynomial of fuzzy graphs. Therefore, in this paper, we consider the chromatic polynomial in fuzzy graph, called fuzzy chromatic polynomial of fuzzy graph. Based on this, we define the concept of fuzzy chromatic polynomial of a fuzzy graph using -cuts of fuzzy graph. Also, we determine the fuzzy chromatic polynomial for fuzzy graphs with crisp and fuzzy vertex set. To determine the fuzzy chromatic polynomial, the classical method for computing the chromatic polynomial of crisp graph is used. Next, we derive some interesting remarks on fuzzy chromatic polynomial of fuzzy graph with crisp and fuzzy vertices. Further, we prove more elegant results on fuzzy chromatic polynomial of fuzzy graphs. Finally, we study a fuzzy chromatic polynomial for complete fuzzy graphs and fuzzy cycles.

The rest of paper is organized as follows. In Section 2, some basic definitions and elementary concepts of fuzzy set, fuzzy graph, and coloring of fuzzy graphs are reviewed. In Section 3, fuzzy chromatic polynomial of a fuzzy graph using -cut of fuzzy graph is defined. Also, fuzzy chromatic polynomials for fuzzy graphs with crisp and fuzzy vertices are determined. In Section 4, more results on fuzzy chromatic polynomials are proved. Furthermore, fuzzy chromatic polynomial for complete fuzzy graphs and fuzzy cycles are studied. Finally, the paper is concluded in Section 5.

#### 2. Preliminaries

In this section, some basic aspects that are necessary for this paper are included. These preliminaries are given in three subsections.

##### 2.1. Basic Definitions and Concepts on Vertex Coloring and Chromatic Polynomial

In this subsection some basic definitions and concepts of vertex coloring and chromatic polynomials are reviewed [30, 31].

*Definition 1. *Let be a graph. A vertex-coloring of is an assignment of a color to each of the vertices of in such a way that adjacent vertices are assigned different colors. If the colors are chosen from a set of colors, then the vertex-coloring is called a -vertex-coloring*, *abbreviated to -coloring, whether or not all colors are used.

*Definition 2. *If has a -coloring, then is said to be -colorable.

*Definition 3. *The smallest , such that is -colorable, is called the chromatic number of , denoted by

*Definition 4. *Let be a simple graph. The chromatic polynomial of is the number of ways we can achieve a proper coloring on the vertices of with the given colors and it is denoted by . It is a monic polynomial in with integer coefficients, whose degree is the number of vertices of .

If we are given an arbitrary simple graph, it is usually difficult to obtain its chromatic polynomial by examining the structure of a graph (by inspection). The following theorem gives us a systematic method for obtaining the chromatic polynomial of a simple graph in terms of the chromatic polynomial of null graphs.

Theorem 5. *Let be a simple graph, and let and be the graphs obtained from by deleting and contracting an edge . Then*

##### 2.2. Basic Definitions on Fuzzy Set and Fuzzy Graphs

In this subsection, some basic definitions on fuzzy set and fuzzy graphs are reviewed [1, 9, 19, 32].

*Definition 6. *A fuzzy set defined on a nonempty set is the family, where is the membership function. In classical fuzzy set theory the set* I *is usually defined as the interval such thatIt takes any intermediate value between 0 and 1 represents the degree in which . The set could be discrete set of the form where indicates that the degree of membership of* x* to* A* is lower than the degree of membership of .

*Definition 7. *-cut set of fuzzy set is defined as is made up of members whose membership is not less than , . -cut set of fuzzy set is crisp set.

*Definition 8. *A fuzzy graph is an algebraic structure of nonempty set together with a pair of functions and such that for all , and is a symmetric fuzzy relation on . Here and represent the membership values of the vertex and of the edge in , respectively.

In this paper, we denote and . Here, we considered fuzzy graph is simple (with no loops and parallel edges), finite, and undirected. is reflective (that is, , for all ) and symmetric (that is, , for all

Note that a fuzzy graph is a generalization of crisp graph in which for all and if and if . So, all the crisp graphs are fuzzy graphs but all fuzzy graphs are not crisp graphs.

*Definition 9. *The fuzzy graph is called a fuzzy subgraph of* G *if for each two elements u, we have and

*Definition 10. *The fuzzy graph is called connected if, for every two elements u, there exists a sequence of elements such that , , and , .

*Definition 11. *For any fuzzy graph , let with elements. Now assume such that . The sequence and the set are called the fundamental sequence and the fundamental set (or level set) of , respectively.

*Definition 12. *The underlying crisp graph of a fuzzy graph is such that and

*Definition 13. *For , -cut graph of a fuzzy graph is a crisp graph such that and . It is obvious that a fuzzy graph will have a finite number of different *α*-cuts.

##### 2.3. Basic Definitions in Fuzzy Coloring of Fuzzy Graphs

The concept of chromatic number of fuzzy graph was introduced by Munoz et al. [22]. The authors considered fuzzy graphs with crisp vertex set, that is, fuzzy graphs, for which for all and edges with membership degree in

*Definition 14 (see [22]). *If is such a fuzzy graph where and is a fuzzy number on the set of all subsets of , assume , where is the fundamental set (level set) of . For each , denote the crisp graph , where and denote the chromatic number of crisp graph .

By this definition the chromatic number of fuzzy graphs G is the fuzzy number , where and

Later Eslahchi and Onagh [21] introduced fuzzy vertex coloring of fuzzy graph. They defined fuzzy chromatic number as the least value of for which the fuzzy graph has -fuzzy coloring as follows.

*Definition 15 (see [21]). *A family of fuzzy sets on a set is called a -fuzzy coloring of if(i),(ii),(iii)for every strong edge (i.e., ) of , (

*Definition 16 (see [21]). *The minimum number for which there exists a -fuzzy coloring is called the fuzzy chromatic number of , denoted as

Incorporating the features of the above two definitions, Kishore and Sunitha [23] modified the chromatic number of fuzzy graph as follows.

*Definition 17 (see [23]). *For each , denote the crisp graph and denote the chromatic number of crisp graph The chromatic number of fuzzy graph is the number .

#### 3. Fuzzy Chromatic Polynomial of a Fuzzy Graph

A fuzzy chromatic polynomial is a polynomial which is associated with the fuzzy coloring of fuzzy graphs. Therefore, chromatic polynomial in fuzzy graph is called fuzzy chromatic polynomial of fuzzy graph. In this section, we define the concept of fuzzy chromatic polynomial of a fuzzy graph based on -cuts of fuzzy graph which are crisp graphs. Furthermore, we determine the fuzzy chromatic polynomials for some fuzzy graphs with crisp and fuzzy vertices.

Crisp vertex coloring and the chromatic polynomial of -cut graph of the fuzzy graph are defined as follows.

*Definition 18. *Let be a fuzzy graph and denote -cut graph of the fuzzy graph which is a crisp graph , A function , is called a -coloring (crisp vertex coloring) of if whenever the vertices and are adjacent in

*Definition 19. *The number of distinct k-coloring on the vertices of is called the chromatic polynomial of . It is denoted by .

Let be the family of -cuts sets of , where the -cut of a fuzzy graph is the crisp graph Hence, any crisp k-coloring can be defined on the vertex set of . The k-coloring function of the fuzzy graph is defined through this sequence. For each , let the fuzzy chromatic polynomial of be defined through a monotone family of sets.

Fuzzy chromatic polynomial of a fuzzy graph is defined as follows.

*Definition 20. *Let be a fuzzy graph. The fuzzy chromatic polynomial of is defined as the chromatic polynomial of its crisp graphs , for . It is denoted by .

That is, , .

##### 3.1. Fuzzy Chromatic Polynomial of Fuzzy Graph with Crisp Vertices

In this subsection, we present fuzzy chromatic polynomial of fuzzy graphs with crisp vertices and fuzzy edges.

A fuzzy graph with crisp vertices and fuzzy edges, and -cut graph of are defined as follows.

*Definition 21. *A fuzzy graph is defined as a pair such that(1)V is the crisp set of vertices (that is, ;(2)the function is defined by , for all

*Definition 22. *Let be a fuzzy graph. For , -cut graph of the fuzzy graph is defined as the crisp graph , where .

*Example 23. *Consider the fuzzy graph with crisp vertices and fuzzy edges in Figure 1.