The Structural Properties of Central Graph of Sun Graphs

Transcription

1 Volume 116 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v116i12.16 THE ACHROMATIC AND B-CHROMATIC NUMBER OF SUN GRAPH, BARBELL GRAPH AND SOME NAMED GRAPHS ijpam.eu K. P. Thilagavathy 1 and A. Santha 2 1 Assistant Professor, Kumaraguru College of Technology, Coimbatore, Kp_thilagavathy_22@yahoo.com 2 Associate Professor, Kumaraguru College of Technology, Coimbatore, Santha_jsk@yahoo.co.in Abstract In this paper, we discuss the structural properties of central graph of Sun graph and central graph of Barbell graph. Here we find out the achromatic and b-chromatic number of Sun graphs and Barbell graph. We apply the achromatic and b-chromatic colouring to these graphs. In addition to that, we find the achromatic and b- chromatic number of Central graph of some named graphs. Key Words achromatic number, b-chromatic number, Barbell Graph, Central Graph, Sun Graph Mathematics Subject Classification: 05C15 1. Introduction Let be a finite un directional graph with no loops and multiple edges. The central graph of a graph is obtained by subdividing each edge of non adjacent vertices of. exactly once and joining all the The Sun graph is a graph on vertices consisting of a central complete graph with an outer ring of vertices, each of which is joined to both end points of the closest outer edge of the central core. The - barbell graph by connecting two copies of a complete graph is obtained by a bridge. Consider a simple graph, the greatest integer such that has a proper colouring and in each colour class there is a representative vertex which is adjacent to at least one vertex 147

2 in every other colour class, is called the b-chromatic number of and is represented by.this colouring concept, called b- colouring [5] was introduced by Irwing & Manlove in An achromatic colouring is a proper vertex colouring such that each pair of colours is adjacent by at least one edge. The largest possible number of colours in an achromatic colouring of is called the achromatic number and it is denoted by. The Structural Properties of Central Graph of Sun Graphs 1. The number of vertices in the Sun Graph is 2. The number of edges in the Sun Graph is q 3. The maximum degree in the Sun Graph is 4. The number of vertices in the Central Graph of Sun Graph is 5. For any Sun Graph 6. For any Sun Graph Theorem: 1 The achromatic number of Central graph of Sun Graph is Proof: Let be the Sun graph with vertices. Let { } { } where the vertices of the complete graph are and are the newly added vertices of the graph. Name the vertices of the complete graph in the cyclic order and in the same cyclic order. In, let be the newly introduced vertex on the edge joining and and be the vertex on the edge joining and Let be the vertex on the edge joining and for Also be the vertex on the 148

3 edge joining and for and be the vertex on the edge joining and. Consider the two sets of colours { } and { }.Assign to and to.to make this colouring as achromatic we have to follow the following procedure: Assign the colour to the vertex for Assign a colour to the vertex Assign the colour to the vertex for Assign to the vertex Assign the colour to the vertex for Assign to the vertex Also assign to the vertex,assign to the vertex The remaining vertices in the complete graph can be given any arbitrary colours in the colour set. By this construction, the above said colouring is achromatic and it is the maximal one. Hence. Example Figure-1 Theorem: 2 149

4 The b-chromatic number of Central graph of Sun Graph is. Observation The b-chromatic number of Central graph of Sun Graph is Example Figure-2 The Structural Properties of Central Graph of Barbell Graphs 1. The number of vertices in the Barbell Graphs is 2. The number of edges in the Barbell Graphs is q 3. The maximum degree in the Barbell Graphs is 4. The number of vertices in the Central Graph of Barbell Graphs is 5. The number of edges in the Central Graph of Barbell Graphs is. 6. For any Barbell Graph 7. For any Barbell Graph Theorem: 3 The achromatic number of Central graph of Barbell Graph is Proof: 150

5 Let be the barbell graph which is made up of two complete graphs connected by an edge. Let { } and { }.Name the first complete graph by in the anti clock wise order. Then name the second complete graph by the name in the clockwise order. Let be the vertices of the edge connecting the two complete graphs. In a vertex in the set is not adjacent to any other vertex in but it is adjacent to all the vertices in the set is an exception. It is adjacent to all vertices in other than Consider the two set of colours { } and { } For assign to and assign to To make this colouring as achromatic we have to follow the following colouring procedure: For assign to and assign to and to For assign to and assign to and to For except assign a colour to and for assign a colour For except assign a colour to and for assign a colour We know that [8] the achromatic number of central graph of any simple graph with is less than or equal to That is, But it is not possible to assign Hence we can conclude that By this construction it is the maximal one. Theorem: 4 The b-chromatic number of Central graph of Barbell Graph is 151

6 Example Figure-3 The Achromatic and b-chromatic number of some named Graphs Franklin Graph The Franklin graph is the 12- vertex cubic graph. For the Franklin graph, For the Franklin graph, Groetzch graph The Grotzch graph is a triangle free graph with 11 vertices, 20 edges and chromatic number 4. The Grotzch graph is smallest triangular free graph. For the Grotzsch graph, Figure-4 For the Grotzsch graph, Mobius graph 152

7 The unique cubic symmetric graph on 16 vertices it is 24 edges, girth 6, diameter 4, chromatic number 2and is non planar but Hamiltonian. For the Mobius graph, For the Mobius graph, Conclusion In this research paper, we compare the structural properties of Sun graph and Barbell graph. Here we find out the achromatic and b-chromatic number of central graphs of Sun graph and Barbell graph. The achromatic number of central graphs of Sun graph and Barbell graphs are equal to and the b-chromatic number of central graph of Sun graph and Barbell graph are equal to Along with this we studied the achromatic and the b-chromatic number of Cubic Symmetry graph, Franklin graph, Grotzch graph and Mobius graphs. References [1] Anitha, R. and Lekshmi, R. S, 2008, "N-Sun Decomposition of Complete, Complete Bipartite and Some Harary Graphs." Int. J. Math. Sci.2, [2] Brice Effantin and Hamammache Kheddouchi, 2003 The b- chromatic number of some power graphs, Discrete Mathematics and Theoretical Computer Science, [3] Gallian, J. 2007,"Dynamic Survey of Graph Labeling." Elec. J. Combin. 14, No. DS6, Jan. 3. [4] Ghosh, A.; Boyd, S.; and Saberi, A., July 24-28,2006. "Minimizing Effective Resistance of a Graph." Proc. 17th Internat. Sympos. Math. Th. Network and Systems, Kyoto, Japan, pp

8 [5] Irving, R.W and Manlove, D.F, 1999, The b-chromatic number of a graph, Discrete Applied Mathematics 91(1-3) [6] Jonathan gross, Jay yellan, 2004, Handbook of graph theory CRC press, New York. [7] Marko Jakovac, Sandi Klavzar, 2010, The b-chromatic number of cubic Graphs, Graphs and Combinatories 26: [8] Thilagavathi, K. Roopesh, N. 2009, Generalisation of Achromatic colouring of Central graphs Electronic Notes in Discrete Mathematics 33, [9] Thilagavathi, K. Thilagavathy, K.P. Roopesh, N. 2009, The achromatic colouring of graphs Electronic Notes in Discrete Mathematics 33, [10] Thilagavathy, K.P, Santha. A, 2016 A Note on Achromatic and b-chromatic number of graphs International Journal of Applied Mathematics and Statistics,53(1), [11] Vernold Vivin, J. Thilagavathi, K, 2006, On Harmonious colouring of Central Graphs, Far East. Math. Sci (FJMS), 2,

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