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Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749 Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749 Graph colouring problem applied in genetic algorithm Malathi R Assistant Professor, Dept of Mathematics, Scsvmv University, Enathur, Kanchipuram, Tamil Nadu 631561, India, Email Id: [email protected] Publication History Received: 06 January 2015 Accepted: 05 February 2015 Published: 18 February 2015 Citation Malathi R. Graph colouring problem applied in genetic algorithm. Indian Journal of Science, 2015, 13(38), 37-41 ABSTRACT In this paper we present a hybrid technique that applies a genetic algorithm followed by wisdom of artificial crowds that approach to solve the graph-coloring problem. The genetic algorithm described here, utilizes more than one parent selection and mutation methods depending u p on the state of fitness of its best solution. This results in shifting the solution to the global optimum, more quickly than using a single parent selection or mutation method. The algorithm is tested against the standard DIMACS benchmark tests while, limiting the number of usable colors to the chromatic numbers. The proposed algorithm succeeded to solve the sample data set and even outperformed a recent approach in terms of the minimum number of colors needed to color some of the graphs. The Graph Coloring Problem (GCP) is also known as complete problem. Graph coloring also includes vertex coloring and edge coloring. However, mostly the term graph coloring refers to vertex coloring rather than edge coloring. Given a number of vertices, which form a connected graph, the objective is that to color each vertex so that if two vertices are connected in the graph (i.e. adjacent) they will be colored with different colors. Moreover, the number of different colors that is used to color the vertices is limited and a secondary objective is that to find the minimum number of different colors needed to color a certain graph without violating the adjacency constraint. That number for a given graph (G) is also known as Chromatic Number (χ(G)) (Isabel Méndez Díaz and Paula Zabala 1999). If k = {1, 2, 3...} and P(G, k) is the possible number of solutions for coloring the graph G with k colors, then χ(G) = min( k : P(G,k)>0) ……… (1) Graph coloring problems are very interesting problems from the theoretical standpoint since they are a class of NP complete problems that also belongs to Constraint Satisfaction Problems (CSPs). The practical applications of GCP (Graph Coloring Problems) include but are not limited to: Map coloring (B. H. Gwee, M. H. Lim and J. S.Ho 1993) Scheduling (Daniel Marx and D Aniel Marx 2004) Radio Frequency Assignment (W. K. Hale 1980;S. Singha, T. Bhattacharya and S. R. B. Chaudhuri 2008) Register allocation (Wu Shengning and Li Sikun 2007) Pattern Matching Sudoku 37 Malathi, Page Graph colouring problem applied in genetic algorithm, Indian journal of Science, 2015, 13(38), 37-41, www.discovery.org.in http://www.discovery.org.in/ijs.htm © 2015 Discovery Publication. All Rights Reserved In this paper we demonstrate the use of genetic algorithms to solve the graph-coloring problem while strictly adhering to the usage of not more than the number of colors equal to the chromatic index to color the various test graphs. Genetic algorithms and evolutionary approaches have been used extensively for the solutions in the Graph Coloring Problem and its applications. The concept of utilizing a crowd of individuals for solving NP complete problems has also been the topic in various papers. Most notably the Wisdom of Crowds concept has been used in solving the Traveling Salesman Problem (Sheng Kung Michael Yi, et al. 2010b) as well as for the Minimum Spanning Tree Problem (Sheng Kung Michael Yi, et al. 2010a). In this paper we attempt to add-on or increase the solution produced by the genetic algorithm utilizing an artificial crowd. Graph or network: A collection of n links connected by m edges is known as graph or network. A network can be directed to the edges of point in one direction. Edges go in both directions is undirected. The edge can join more than two vertices together is well – known as hyper graph. List coloring: Generally, for a function (f) assigning a positive integer f(v) to each vertex(v ), a graph G is f-choosable (or f-list- colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to each vertex (v). In particular, if for all vertices (v), f-choosability corresponds to k-choosability. List edge coloring: In mathematics, list edge-coloring is a type of graph coloring that joins list coloring and edge coloring. An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors, for each edge. A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper only if no two adjacent edges receive the same color. A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its fundamental graph and that provides at least k allowed colors for each edge of G has a proper coloring. The edge choosability, or list edge color ability list edge chromatic number, or list chromatic index, ch′(G) of graph (G) is the least number k such that G is k-edge-choosable Total coloring: In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent edges and no edge and its end vertices are assigned the same color. The total chromatic number χ″(G) of a graph G is the least number of colors needed in any total coloring of G. The total graph T = T(G) of a graph (G) is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T their corresponding elements are either adjacent or incident in G. Then total coloring becomes a proper vertex coloring of the total graph. Some properties of χ″(G): 1. χ″(G) ≥ Δ( G) + 1. 2. χ″(G) ≤ Δ( G) + 1026. 3. χ″(G) ≤ Δ( G) + 8 ln8(Δ(G)) for sufficiently large Δ(G). 4. χ″(G) ≤ ch′(G) + 2. Here Δ(G) is the maximum degree; and ch′(G), the edge choosability. Total coloring occurs naturally since it is simply a combination of vertex and edge colorings. The next step is to look for any Brooks-typed orVizing-typed upper bound on the entire chromatic number in terms of maximum degree. It turns out that the total coloring version of maximum degree upper bound is a difficult problem and has evaded mathematicians for 40 years. Harmonious colouring In graph theory, a harmonious coloring is a proper vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number χH(G) of a graph G is the least number of colors needed for any harmonious coloring of G. Every graph has a harmonious coloring, since it suffices to allot every vertex a distinct color; thus χH(G) ≤ |V(G)|. They slightly exist graphs G with χH(G) > χ(G) (where χ is the chromatic number); one example is the path of length 2, which can be 2-colored but has no harmonious coloring with 2 colors. Some properties of χH(G): χH(Tk,3) = (3/2)(k+1) , where Tk,3 is the complete k-ary tree with 3 levels (Mitchem 1989). Harmonious coloring was first proposed by Harary and Plantholt (1982). Still very little is known about it. ⌈ ⌉ 38 Malathi, Page Graph colouring problem applied in genetic algorithm, Indian journal of Science, 2015, 13(38), 37-41, www.discovery.org.in http://www.discovery.org.in/ijs.htm © 2015 Discovery Publication. All Rights Reserved Complete colouring: In graph theory, complete coloring is the reverse of harmonious coloring in the sense that it is a vertex coloring in which every pair of colors appears on at least one pair of adjacent vertices. Equivalently, a complete coloring is least in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number ψ(G) of a graph G is the highest number of colors possible in any complete coloring of G. Extract colouring: In graph theory, an exact coloring is a proper vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. In essence, an exact coloring is a coloring that is together harmonious and complete. Graphs that confess exact colorings have been classified. Cyclic colouring: In graph theory, an acyclic coloring is a proper vertex coloring in which every 2-chromatic sub – graph is acyclic. The acyclic chromatic number A(G) of a graph (G) is the minimum number of colors needed in any acyclic coloring of G. Acyclic coloring is often associated with graphs fixed on non-plane surfaces. Star colouring: In graph-theoretic mathematics, a star coloring of a graph G is a proper vertex coloring in which every path on four vertices uses at least three separate colors. Equivalently, in a star coloring, the induced sub - graphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number of G is the minimum number of colors needed to star color G.
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