2- Dominator Coloring for Various Graphs in Graph Theory

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2015): 6.391 2- Dominator Coloring for Various Graphs in Graph Theory A. Sangeetha Devi1, M. M. Shanmugapriya2 1Research Scholar, Department of Mathematics, Karpagam University, Coimbatore-21, India . 2Assistant Professor, Department of Mathematics, Karpagam University, Coimbatore-21, India Abstract: Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graphG dominates at least 2-color class. In this paper, as an extension of Dominator coloring such that various graph using 2- dominator coloring has been discussed. Keywords: 2-Dominator Coloring, Barbell Graph, Star Graph, Banana Tree, Wheel Graph 1. Introduction of all the pendent vertices of G by „n‟ different colors. Introduce two new colors „퐶푖 „, „퐶푗 ‟alternatively to the In graph theory, coloring and dominating are two important subdividing vertices of G, C0 will dominates „퐶푖 „, „퐶푗 ‟. Also areas which have been extensively studied. The fundamental all the pendent verticies of G dominates at least two color parameter in the theory of graph coloring is the chromatic class. i.e, 1+n+2 ≤ 푛 + 3. Therefore χ푑,2 {C [ 푆1,푛 ] }≤ 푛 + number χ (G) of a graph G which is defined to be the 3 , n≥ 3. minimum number of colors required to color the vertices of G in such a way that no two adjacent vertices receive the Theorem 2.2 same color. If χ (G) = k, we say that G is k-chromatic. Let 푆 be a star graph, then χ (푆 ) =2, n 1,푛 푑,2 1,푛 .2 A dominating set S is a subset of the vertices in a graph such that every vertex in the graph either belongs to S or has a Proof: neighbor in S. The domination number is the order of a Let 푆1,푛 be a star graph by 2 dominator coloring and from minimum dominating set. Given a graph G and an integer k, theorem 3.2 and 3.7[9], the assignment of colors to the finding a dominating set of order k is NP-complete on vertices of G, May dominates itself and it forms a color arbitrary graphs [2], [4]. class. Thus χ푑,2(푆1,푛 ) =2. Graph coloring is used as a model for a vast number of Theorem 2.3 practical problems involving allocation of scarce resources Let G=B ( 퐾푛 , 퐾푛 ) where n=3, 4, 5, . .. Then χ푑,2 [B (e.g., scheduling problems), and has played a key role in the (퐾푛 , 퐾푛 ) ]=n. development of graph theory and, more generally, discrete mathematics and combinatorial optimization. A graph has a Proof: dominator coloring if it has a proper coloring in which each Let G= B (퐾푛 , 퐾푛 ) be a Barbell graph. Now divide the vertex of the graph dominates every vertex of some color vertices of G be U and V. Let U= {푢0 ,푢1 , 푢2, … ., 푢푛−1 } class. The dominator chromatic number χ (G) is the d and V = {푣0, 푣1,푣2,…..푣푛−1} respectively. By definition of minimum number of color classes in a dominator coloring of Barbell graph, G is obtained by connecting two complete a graph G. A χd(G) - coloring of G is any dominator coloring graphs 퐾푛 by 푎푛 bridge 푒푖 . Using theorem 1.2 [2] and 3.1 with χd(G) colors. Our study of this was motivated by [9] and [3],2-dominator coloring in any complete graph all the [7]. vertices must contain at least n colors. Let the color vertex 푢0 by 푐0 ,푢1 by 푐1, 푢2 by 푐2, … , 푢푛 by 푐푛−1 . Similarly, color of 2. Various Kinds of Graphs in Graph Theory the vertex 푣0 by 푐0 ,푣1 by 푐1, 푣2 by 푐2, … , 푣푛 by 푐푛−1 . Thus a bridge „푒푖 ‟ between U and V will receive the color „푐0 푐푛−1 .‟. Theorem 2.1 Hence no two adjacent vertices have same color. Also every Let 푆1,푛 be a star graph, C vertex of G dominates at least two color classes. 푆 ] 푏푒푎푐푒푛푡푟푎푙푔푟푎푝푕표푓푠푡푎푟 푆 ] [ 1,푛 graph, then χ푑,2 {C [ 1,푛 } i.e. χ푑,2 [B (퐾푛 , 퐾푛 ) ] = n, n=3,4,5,…. n+2,n≥ 3. Theorem 2.4 Proof: Let G=푃2 푃푛 , n≥ 2 , then . χ 퐺 = 4. 푈 푣 푣 푣 푣 푑,2 Let 0 be the root vertex of G. Let { 0, 1, 2,….. 푛−1} be G the remaining vertices of . By definition of central graph Proof: C(G), subdivide the each edge exactly once and joining all Let G = 푃 푃 be a ladder graph , n≥ 2 (i.e. two rails or the non-adjacentvertices of G. Let푤 ,푤 , 푤 , … … 푤 be 2 푛 0 1 2 푛−1 paths connected by rungs). Now color the vertices in path 푃 the subdividing vertices of G. By definition of 2- dominator 1 by two different colors 푐0 , 푐1 alternatively. Similarly color. coloring, assign color 퐶0 to the root vertices푢0 . Now color The vertices in path 푃2 by another new two different colors Volume 5 Issue 4, April 2016 www.ijsr.net Paper ID: 16041604 Licensed Under Creative Commons Attribution CC BY 1414 International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2015): 6.391 푐2 , 푐3 alternatively. Therefore the every rung 푃푛 between the C1 and C2 in G. By definition of 2-Dominator coloring the paths 푃1 and 푃2 will receive the color 푐0 , 푐2 and 푐1 , 푐3 vertices of G must dominates at least two color classes. alternatively. Hence by definition of 2 dominator coloring Hence χ푑,2 (퐺3 ) = 4, n = 4,5,6,….. χ푑,2(푃2푃푛 ) = 4,n ≥ 2. Theorem 2.9 Theorem 2.5 If G=푊푛 be a wheel graph then 3, 푛 = 6,8,10, … Let G=퐵푚,푛 , m,n≥ 2 be a banana tree then χ푑,2 퐵푚,푛 = 2 χ푑,2(푊푛 ) = 4, 푛 = 7,9,11, … Proof: Proof: Let 퐵 be a banana tree of order m,n≥ 2 ie (n,m)- banana 푚,푛 Let G=푊 be a wheel graph. tree is a graph created by connecting a leaf of each of m 푛 Let u1 be the root vertex of G. copies of n-star graph with a pendent vertex and it is Let 푣 ,푣 ,…..푣 be the main vertex of G. different from all stars. Thus from theorem 3.2 and 3.7[9], 1 2 푛 퐵 = 2. χ푑,2 푚,푛 2- Dominator coloring of G as follows. Case 1: χ푑,2 푊푛 = 3, 푛 = 6,8,10, … Theorem 2.6 First color 푐0 to the root vertex 푢1 . Assign color „퐶1 , 퐶2 ‟ If G=퐹푛,푘 , n,k ≥ 2, then χ 퐺 = 2 푑,2 alternatively to the remaining verticies of G. i.e 푣1푣2 receive color퐶1퐶2 , 푣2푣3 receive color 퐶2퐶1 , 푣3푣4 will receive 퐶1퐶 Proof: and so on. By coloring it forms a clique. Hence every vertex Let G=퐹푛,푘 , n,k ≥ 2, be a fire cracker graph i.e 퐹푛,푘 is a must dominate at least two color classes. Thus χ푑,2 푊푛 = graph connecting n, k- states by an leaf. By definition of 2 3, 푛 = 6,8,10, … dominator coloring and by use of theorem 3.2 and 3.7[9], Case 2: χ푑,2 푊푛 = 4, 푛 = 7,9,11, …. every vertex may dominates its own class. Thusχ푑,2 퐹푛,푘 = Assign color 퐶0 to the root vertex 푢1 . Now the 2. colors 퐶1 퐶2 , 퐶3 will be assigned alternatively to the remaining vertices of G. i.e.v1v2 by C1C2, v2v3 by C2C3, v3v4 Theorem 2.7 byC3C1, v4v5by C1C2,……,,and so on.By coloring it forms a 푚 If G=퐷푛 , n be the number of vertex m be the copies of clique while assigning the color in the above alternative 푚 sequence, at last there may or may not remains some vertices complete graph, n,m≥ 3 , then, χ푑,2[퐷푛 ] = 푛. of G, it depends up on „n‟ suppose the vertices of G remains, Proof to color the remaining vertices of G using the above same 푚 three colors 푐 푐 , 푐 colored in any order, but there should Let G= 퐷 be a wind mill graph and it is obtained of 1 2 3 푛 not any two adjacent vertices receive the same color. Hence 퐾 connecting m copies of 푛 with the common vertex. By the every vertex must dominate at least two color classes. Thus definition of 2- dominator coloring from theorem 1.2 and 푚 χ푑,2 푊푛 = 4, 푛 = 7,9,11, …. 3.1[3]퐷푛 needs at least n colors and no two adjacent 푚 vertices will receive same color. Thus χ푑,2 [ 퐷푛 ] = 푛 , Theorem 2.10: ≥ 3,4,5, … n For any prism graph 푌2,푛 , χ푑,2(푌2,푛 ) ≤ 푛 , n=3,4,5,….. Theorem 2.8 Proof: G 퐺 퐺 = 4, 푛 = 3,4,5,7, … If = 푛 be any gear graph then χ푑,2 Let G = 푌2,푛 be a prism graph. Let G = 푈 ∪ 푉 . Let U= {푢0 ,푢1 , 푢2, … ., 푢푛−1 } be the inner vertices of G. V = {푣0, Proof: 푣1,푣2,…..푣푛−1} be the outer vertices of G respectively. Let Let G=퐺푛 be a gear graph or bipartite wheel graph. E(U)= {푢푖 푢푖+1; 0≤ 푖 ≤ 푛 − 2} and E(V)= { 푣푖 푣푖+1; 0≤ 푖 ≤ Let u1 be the root vertex of G.

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