Dominator Coloring of Prism Graph

Transcription

1 Applied Mathematical Sciences, Vol. 9, 0, no. 38, HIKARI Ltd, Dominator Coloring of Prism Graph T. Manjula Department of Mathematics, Sathyabama University, Chennai, India R. Rajeswari Department of Mathematics, Sathyabama University, Chennai, India Copyright 0 T. Manjula and R. Rajeswari. 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 A graph has a dominator coloring if it has a proper coloring in which each vertex of the graph dominates every vertex of some color class. The dominator chromatic number χd (G) is the minimum number of colors required for a dominator coloring of G. In this paper the dominator chromatic number for Prism graph is studied and also the relation between dominator chromatic number, domination number and chromatic number is shown. Keywords: Coloring, Domination, Dominator Coloring Introduction Let G= (V, E) be a graph such that V is the vertex set and E is the edge set. A dominating set S is a subset of the vertex set V of graph G such that every vertex in the graph either belongs to S or adjacent to S. The domination number γ (G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of a graph to a set of colors such that any two adjacent vertices have different colors. A subset of vertices colored with the same color is called a color class. The chromatic number is the minimum number of colors needed in a proper coloring of a graph and is denoted by χ (G). A dominator coloring of a graph G is a proper coloring of graph such that every vertex of V dominates all vertices of at least one color class (possibly its own

2 890 T. Manjula and R. Rajeswari class). i.e., it is coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class and this concept was introduced by Ralucca Michelle Gera in 00 []. The dominator chromatic number χd (G) is the minimum number of color classes in a dominator coloring of G. The relation between dominator chromatic number, chromatic number and domination number of some classes of graphs were studied in [], [3]. The dominator coloring of bipartite graph, star and double star graphs, central and middle graphs, fan, double fan, helm graphs etc. were also studied in various papers [], [], [7], [8]. In this paper the dominator chromatic number for Prism graph is studied and also the relation between dominator chromatic number, domination number and chromatic number is shown. A graph corresponding to the skeleton of an n-prism is called a prism graph and is denoted by Y n. A prism graph also called as circular ladder graph, has n vertices and 3n edges. Prism graphs are both planar and polyhedral. It is equivalent to the generalized Petersen graph P n, and the Cayley graph of the dihedral group D n. Dominator Coloring of Prism graph Theorem.: For a prism graph Y n, n 9, χ d (Y n ) = n + Proof: The vertex set of the prism graph is given by V = {v i i n} and the edge set is defined ase = E E E 3 E where E = { v i v i+ ; i n }, E = v n v, E 3 = v n v n+ and E = { v i v n+i+ ; i n }. The following procedure gives a dominator coloring of prism graph Y n. Let the adjacent vertices of v, v namely {v, v n, v n+ }, {v, v 3, v n+3 } be assigned color and the color respectively. The vertex v is given color 3. Let the vertices v i, v n+i for i n be given color i. The vertices v n, v n+, v n+ are given colors n, n, n + respectively. Let the adjacent vertices of v, v namely {v, v n, v n+ }, {v, v 3, v n+3 } be assigned color and the color respectively. The vertex v is given color 3. Let the vertices v i, v n+i for i n be given color i. The vertices v n, v n+, v n+ are given colors n, n, n + respectively. Then the vertices v i, i n, dominate color class i. The vertices v n+, v n+ dominate color class n and v n+ 3, v n+, v n+ dominate color class n +. The vertex v n+i for i n, dominates color class i.

3 Dominator coloring of prism graph 89 Thus the minimum number of colors needed for dominator coloring of a prism graph Y n for n 9 is n +. i.e., the dominator chromatic number is χ d (Y n ) = n +, for n 9. Illustration.: Figure shows dominator coloring of prism graph Y. v 7 v 8 v 0 3 v v v v v 3 v 0 8 v v 3 v 8 v v v 9 v8 v 7 7 v v v 3 v v 7 v v 7 3 v 8 v 9 v 0 v Figure The vertices v i, i, dominate color class i. The vertices v, v dominate color class and v 7, v 8, v 9 dominate color class. For i, the vertex v +i dominates color class i. Thus χ d (Y ) =. Observation.: The dominator chromatic number of a prism graph Y n, 3 n 8, n, is n i.e., χ d (Y n ) = n

4 89 T. Manjula and R. Rajeswari Illustration.: Figure shows the dominator coloring of prism graph Y. v8 v9 v v v8 v v3 v0 3 v v 3 v v Figure The vertices v, v, v dominate color class,, respectively. The vertices v 3, v dominate color class, 3 respectively. The vertex v dominate color class. The vertex v dominate color class the vertices v 7, v 8 dominate color class and the vertices v 9, v 0, v dominate color class. Thus χ d (Y ) =. Observation.3: The dominator coloring of a prism graph Y n when n = is. i.e., χd (Y ) =. Illustration.3: Figure 3 shows the dominator coloring of prism graph Y. The vertices v i, i, dominate color class i. The vertices v, v 7 dominate color class and the vertices v +i, 3 i dominate color class. Thus the minimum number of colors needed for dominator coloring of Y is. i.e., χd (Y ) =.

5 Dominator coloring of prism graph 893 v7 v v v v v8 3 v v3 v0 v9 Figure 3 Theorem.: For a prism graph Y n, n 3, χ d (Y n ) > γ(y n ) Proof: The dominator chromatic number for a prism graph Y n is given by n + for n =, and n 9 (Y n ) = { n for n and 3 n 8 χ d The domination number for a prism graph Y n is given by [] n if n (mod ) γ(y n ) = { n + if n ( mod ) n+ if n (mod ) Hence χ d (Y n ) > γ(y n ), n 3 Theorem.: For a prism graph Y n, n 3, χ d (Y n ) χ(y n ) Proof: The dominator chromatic number for a prism graph Y n is given by n + for n =, and n 9 (Y n ) = { n for n and 3 n 8 χ d The chromatic number for a prism graph Y n is given by [9] χ(y n ) = when n is even { 3 when n is odd

6 89 T. Manjula and R. Rajeswari Thus for n > 3, χ d (Y n ) > χ(y n ). For n = 3, χ d (Y n ) = χ(y n ) = 3. Thus the bound is sharp. Hence χ d (Y n ) χ(y n ). References [] S Arumugam, Jay Bagga and K Raja Chandrasekar. On dominator colorings graphs, Proc. Indian Acad. Sci. (Math. Sci.) Vol., No., November 0, pp. 7, Indian Academy of Sciences. [] R. Gera, S Horton, C. Rasmussen, Dominator Colorings and Safe Clique Partitions, Congressus Numerantium (00). [3] Gera R M. On dominator coloring in graphs, Graph Theory Notes N.Y. LII (007) 30. [] Gera R. On the dominator colorings in bipartite graphs, IEEE Computer Society (007) [] A. D. Jumani, L. Chand, Domination Number of Prism over Cycle, Sindh Univ. Res. Jour. (Sci. Ser.) Vol. () (0). [] Kavitha, N G David, Dominator Coloring on Star and Double Star Graph Families, International Journal of Computer Applications: -, June 0. [7] K. Kavitha, N.G. David, Dominator Coloring of Central Graphs International Journal of Computer Applications ( ) Vol. No., Aug 0. [8] K. Kavitha & N. G. David, DOMINATOR COLORING OF SOME CLASSES OF GRAPHS, International Journal of Mathematical Archive- 3 (), Nov. 0. [9] S. Sudha, G. M. Raja, Equitable Coloring of Prisms and The Generalized Petersen graphs, International Journal of Research in Engineering & Technology, Vol., Issue, Feb 0, 0-, Impact Journals. Received: February 7, 0; Published: March 0, 0