International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019

Power Dominator Chromatic Number for some Special Graphs

A. Uma Maheswari, Bala Samuvel J.

 Definition 1: Proper Coloring Abstract: Let G = (V, E) be a finite, connected, undirected with A proper coloring [4] of a graph G is an assignment of colors no loops, multiple edges graph. Then the power dominator to the vertices of the graph such that no two adjacent vertices coloring of G is a proper coloring of G, such that each of G have the same color and the chromatic number χ(G) of the power dominates every vertex of some color class. The minimum graph G is the minimum number of colors needed in a proper number of color classes in a power dominator coloring of the coloring of G. graph, is the power dominator chromatic number . Here Definition 2: Dominator Coloring we study the power dominator chromatic number for some special A dominator coloring [[5],[6],[7],[8],[9]] of a graph is a graphs such as Bull Graph, Star Graph, Wheel Graph, Helm proper coloring such that each vertex dominates every vertex graph with the help of induction method and Fan Graph. Suitable in at least one color class consisting of vertices with the same examples are provided to exemplify the results. color. The chromatic number of a graph G is the

Keywords: Bull graph, Coloring, Power dominator coloring, minimum number of colors needed in a dominator coloring of star graph, Wheel graph. G. AMS Mathematics Subject Classification (2010): 05C15, Definition 3: Bull Graph 05C69 A bull graph is the planar undirected graph with 5 vertices and 5 edges constructed by inducting two pendent vertices to I. INTRODUCTION any two vertices of . Definition 4: Star Graph In , Domination and Coloring are two main A star K1,n is a tree with n vertices of degree 1and root vertex has degree n. areas of study in graph theory and these topics have been Definition 5: Wheel Graph [6] extensively explored by considering different variants, (see, A wheel graph is the join of and star graph . [[1],[2],[3]]. Definition 6: Helm Graph [9] While formulating a problem related to electric power The Helm Graph is the graph obtained from a wheel system in graph theoretical terms, Haynes et al. [4] graph by attaching a pendant edge at each vertex of the introduced the concept of power domination. A vertex set S n – cycle. of a graph G is defined as the power dominating set of graph Definition 7: Flower Graph [9] if every vertex and every edge in the graph is monitored by S, A flower graph is the graph obtained from a helm by with following a set of rules for power monitoring system. joining each pendant vertex to the central vertex of the helm. Based on the ideas of coloring and power domination, K.S. Definition 8: Monitoring Set Kumar & N.G. David [[5],[6]] presented a new coloring For a vertex , a monitoring set M(v)[6] is associated variant called power dominator coloring of graph. as follows: In this paper, we consider the graphs that are Step(i): M(v) = N[v], the closed neighborhood on v finite, connected, undirected with no loops and multiple Step(ii) : Add a vertex u to M(v), (which is not in M(v)) edges and find the power dominator chromatic numbers of whenever u has a neighbor w M(v) such that all the some special graphs such as Bull Graph, Star Graph, Wheel neighbors of w other than u, are already in M(v) Graph, Helm graph, and Fan Graph. Step(iii): Repeat Step(ii) if no other vertex could be added to M(v). II. PRELIMINARIES Then we say that v power dominates the vertices in M(v). Note that if a vertex v dominates [13] another vertex u, then v Some of the basic definitions required for this paper are power dominates u but the converse need not be true. recalled below. Definition 9: Power Dominator Coloring The power dominator coloring of G is a proper coloring of G, such that every single vertex of G power dominates all vertices of some color class. The minimum number of color Revised Manuscript Received on October 05, 2019. classes in a power dominator coloring of the graph, is the * Correspondence Author A. Uma Maheswari*, Associate Professor, PG & Research Department power dominator chromatic number. And it is denoted by of Mathematics, Quaid-E-Millath Government College for Women . (Autonomous), Chennai – 600 002, Tamilnadu, India. In Example 1, we explain the concept of power dominator Email: [email protected] Bala Samuvel J, Research Scholar, PG & Research Department of coloring with the graph Fig 1. Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai – 600 002, Tamilnadu, India. Email: [email protected]

Retrieval Number: L34661081219/2019©BEIESP Published By: DOI: 10.35940/ijitee.L3466.1081219 Blue Eyes Intelligence Engineering 3957 & Sciences Publication

Power Dominator Chromatic Number for some Special Graphs

We consider the graph G in Fig. 1. The vertex is colored For the power dominator chromatic number of the star by color 1 . The vertices are assigned graph, the color 2 and the vertices are assigned the Proof color 3 . Every individual vertices of The vertices of the star graph be the root vertex, pendent power dominates the vertex vertices are , 1 . Assign the color to the root with color . And vertex power dominates { by the vertex and color to the pendent vertices , 1 . definition of power dominator coloring. The root vertex power dominates the color class Example 1 , each pendent vertex power dominate color class . This guarantees that the power dominator coloring is proper. The power dominator chromatic number for the star graph Example 3 In Figure 3, let us present the power dominator coloring of star graph

For instance, consider , it power dominates , which power dominate and which in turn power dominate with color . Hence the vertex power dominate the vertex v1 with color . In the similar way we can show that, every vertex in the Fig. 1, can power dominate with color . Therefore, it is clear that, not less than 3 colors will be sufficient to offer a power dominator coloring for G in Fig.1. The result relating the chromatic number, power dominator chromatic number and dominator chromatic number was proved in [11]. Theorem 1 [11]. For any graph G In the Fig. 3 the color classes of the star graph are Remark: For the graph in Fig. 1, Then . Hence . . Theorem 3 III. MAIN RESULTS For any the power dominator chromatic number of the Theorem 1 wheel graph The power dominator coloring for a Bull graph G,

Proof Proof Let graph be the bull graph with 5 vertices. The Case (i) Wheel graph , when is 3 vertices of the graph be . A power dominator Let be the central vertex and the vertices on the cycle are coloring of G is given as, Color the vertices with . Let us color the central vertices with color , vertices and colored with and respectively. and with color , and respectively. And So vertices power dominate and vertices with , color classes power dominates itself by power dominate with . Both the vertices power definition of power dominator coloring and vertex power dominate and power dominate . dominates the color classes The above Thus any vertex of G power dominates some color class. The procedure gives us proper power dominator coloring for the power dominator coloring for a Bull graph G is . wheel graph , when is 3. And the power dominator Example 2 chromatic number for wheel graph is 4. i.e., In Figure 2, the power dominator coloring of bull graph G is . shown. Case (ii) Wheel graph , when is odd number greater than 3 Let be the central vertex and the vertices on the cycle are . Let us color the central vertices with color , all odd indexed vertices with color , all even indexed vertices except with color and vertex is colored by . And color classes power dominates itself by definition of power dominator coloring and vertex power dominates the color classes The above procedure gives us proper In the Fig. 2 the color classes of the bull graph are power dominator coloring for & . Then . the wheel graph , when Theorem 2 is odd. The power dominator

Published By: Retrieval Number: L34661081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3466.1081219 3958 & Sciences Publication International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019 chromatic number for wheel graph is 4. i.e., central vertex and pendent vertices with . color and with color , , and Case (iii) Wheel graph , when is even number greater respectively. And color classes power than 3 dominates itself by definition of power dominator coloring Let be the central vertex and the vertices on the cycle are and vertex power dominates the color classes Let us color the central vertices with Each pendent vertices power color , all odd indexed vertices with dominate the adjacent vertex with the color class color , and all even indexed vertices with . This will ensure proper power dominator color . And color classes power dominates itself by coloring for the helm graph, when is 3. And the power definition of power dominator coloring and vertex power dominator chromatic number for helm graph, when is 4. dominates the color classes . The above , when . procedure gives us proper power dominator coloring for the We assume the result is true for We prove that the wheel graph , when is odd. The power dominator power dominator chromatic number for the helm graph , chromatic number for wheel graph is 4. i.e., . Let be the central vertex and the vertices on the rim are and be the pendent Example 3 vertices adjacent to . Let us color the In Figure 4, the power dominator coloring of wheel graph central vertex and pendent vertices is presented with color , and with color , , ,…, respectively. The color classes power dominates itself by definition of power dominator coloring and a vertex power dominates the color classes Each pendent vertices power dominate the adjacent vertex with the color class . The above procedure provides a proper power dominator coloring. The power dominator chromatic number for the helm graph , . Hence the result. Example 4 In Figure 5, the power dominator coloring of helm graph is presented

In the Fig. 4 the color classes of the wheel graph of are , . Then . Theorem 4 For any the power dominator chromatic number of the Helm graph . Proof We prove this theorem by method of induction. First we prove the theorem for Let be the central vertex and the vertices on the cycle are and be the pendent vertices adjacent to respectively. Let us color the central vertex and pendent vertices with color and with color , and respectively. And color classes power dominates itself by definition In the Fig. 5 the color classes of the helm graph are of power dominator coloring and vertex power dominates , . Then the color classes Each pendent vertices . power dominate the adjacent vertex. This will ensure proper Theorem 5 power dominator coloring for the helm graph, when is 3. For the power dominator chromatic number for the The power dominator chromatic number for helm graph, Flower graph when is 3. i.e., . Next, we prove the theorem for Proof Let be the central vertex and Let be the central vertex and the vertices on the cycle are the vertices on the rim be and be the pendent vertices and adjacent to respectively. Let us color the

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Power Dominator Chromatic Number for some Special Graphs

be the outer vertices adjacent to IV. CONCLUSION . The concept of power dominator coloring relates coloring Case (i) When is odd, problem with power dominating sets in graphs. Computing The vertex is colored by color . the vertices , the power dominator chromatic number for some special on the rim are colored by colors and graphs such as Bull Graph, Star Graph, Wheel Graph, Helm alternatively and assign color to vertex . The graph, and Fan Graph has been the main focus of this paper outstanding vertices are colored with and the results are presented in the table 1. There is the scope and alternatively. The vertex power dominates for finding the power dominator chromatic number for product itself. The vertex , and , 1 graph. power dominates the color class , for, it is adjacent to . Hence in this case the power dominator chromatic number for REFERENCES flower graph , if is odd. 1. C. Kaithavalappil, S. Engoor, and S. Naduvath, “On Equitable Case (ii) When is even Coloring Parameters of Certain Wheel Related Graphs,” no. October, The vertex is colored by color . Assign colors and 2017. 2. R. Gera, “On the Dominator Colorings in Bipartite Graphs,” Discuss. alternatively to the vertices , on the Math. - Graph Theory, vol. 32, no. 4, pp. 677–683, 2012. rim. The outstanding vertices are 3. N. K. Sudev, K. P. Chithra, and J. Kok, “Some Results on the colored by colors and alternatively. the vertex b-Colouring Parameters of Graphs,” pp. 1–11, 2017. 4. J. A. Bondy and U. S. R. Murty, “Graph Theory with Applications,” power dominates itself. Math. Gaz., vol. 62, no. 419, p. 63, 1978. The vertex , and , 1 power 5. S. Arumugam and J. A. Y. Bagga, “On dominator colorings in graphs,” dominates the color class , for, it is adjacent to . The vol. 122, no. 4, pp. 561–571, 2012. 6. S. Arumugam, K. R. Chandrasekar, N. Misra, G. Philip, and S. power dominator chromatic number for flower graph is Saurabh, “Algorithmic aspects of dominator colorings in graphs,” Lect. 3, when is even. i.e., , if is even. Notes Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Example 6: Notes Bioinformatics), vol. 7056 LNCS, no. 1, pp. 19–30, 2011. In Figure 6, let us present the power dominator coloring of 7. R. Gera, C. Rasmussen, and H. Steve, “Dominator Colorings and Safe Partitions,” Congr. Numer, vol. 181, no. 4, pp. 19–32, 2006. flower graph 8. D. K. Syofyan, E. T. Baskoro, and H. Assiyatun, “The Locating-Chromatic Number of Binary Trees,” Procedia Comput. Sci., vol. 74, pp. 79–83, 2015. 9. K. Kavitha and N. G. David, “Dominator Coloring of Some Classes of Graphs,” Int. J. Math. Arch., vol. 3, no. 11, pp. 3954–3957, 2012. 10. T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi, and M. A. Henning, “Domination in Graphs Applied to Electric Power Networks,” SIAM J. Discret. Math., vol. 15, no. 4, pp. 519–529, 2002. 11. K. S. Kumar, N. G. David, and K. G. Subramanian, “Graphs and Power Dominator Colorings,” vol. 11, no. 2, pp. 67–71, 2016. 12. K. S. Kumar, N. G. David, and K. G. Subramanian, “Power Dominator Coloring of Certain Special Kinds of Graphs,” vol. 11, no. 2, pp. 83–88, 2016. 13. T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi, and M. A. Henning, Fundamentals of Domination in Graphs. 1998. 14. P. Suganya and R. M. J. Jothi, “DOMINATOR CHROMATIC NUMBER OF SOME GRAPH CLASSES,” Int. J. Comput. Appl. Math., vol. 12, pp. 458–463, 2017.

AUTHORS PROFILE

Dr.A.Uma Maheswari is an Associate In the Fig. 6 the color classes of the flower graph are Professor, Departement of Mathematics, . , Quaid-E-Millath Govt College for Women, . Then . Chennai-002. She has teaching experience of 26 years. She was CSIR, JRF, SRF and has completed one UGC The results which are investigated are presented in the table research award scheme, one UGC Major project, one UGC Minor project below. and one TANSCHE Minor Research project. She has more than 100 Table 1 international publications. She has chaired a mathematical session at University of Cambridge and presented a paper at Oxford University. She is a recipient of 2 international and 5 national awards. Name of Number of Power Dominator Chromatic the Graph Vertices Number

Bull Graph 5 Bala Samuvel J is a research scholar currently pursuing Ph.D (Part time) under the Supervision of Star Graph n + 1 Dr.A.Uma Maheswari is an Associate Professor, Wheel Department of Mathematics, Quaid-E-Millath Govt College 2n for Women, Chennai-002. Graph

Helm 2n + 1 Graph

Fan Graph 2n + 1

Published By: Retrieval Number: L34661081219/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.L3466.1081219 3960 & Sciences Publication