Achromatic Colouring of the Central Graph of Some Specialgraphs

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-11S, September 2019

K.P.Thilagavathy, A.Santha, G.S. Nandakumar 

Abstract— In this research investigation, the achromatic number of central graph of double wheel graph, wind mill graph andn- anti prism graph have been studied. In addition the structural properties of these graphs have also been studied.

Key Words: Double wheel graph, Wind mill graph, Anti prism graph, achromatic number, b-chromatic number, Central graph Mathematics subject classification: 05C15

I. INTRODUCTION Wind mill graph Consider a simple undirected graph G . To form its central graph C(G) , we introduce a new node on every An anti prism is a semi regular polyhedron constructed with gons and triangles. It is made edge in and join those nodes of that are not up of two gons on the top and bottom separated by a adjacent.The achromatic number was introduced by Harary. ribbon of triangles, with the two gons being offset A proper vertex colouring is said to be achromatic if every by one ribbon. The graph corresponding to the skeleton of pair of colours has at least one edge joining them . The an anti prism is called the anti prism graph and is achromatic number is the maximum number of colours denoted by in an achromatic colouring of . A double wheel graph of size is composed of and It consists of two cycles of size where the vertices of the two cycles are connected to a central root vertex.

Anti prism graph

Structural properties of central graph of double wheel graph

 The number of vertices in Double wheel  The number of edges in  The maximum degree in = . The wind mill graph is obtained by joining copies of the complete graph with a vertex in  The number of vertices in common.

 The number of edges in =

Theorem: 1

For a double wheel graph , the achromatic number of is Proof: Revised Manuscript Received on September 10, 2019. A double wheel graph of size is composed of K.P.Thilagavathy, Assistant Professor, Department of Science and Humanities, Kumaraguru College of Technology, Coimbatore, Tamilnadu, and Consider the two sets of vertices India. A.Santha, Associate Professor, Department of Science and Humanities, Kumaraguru College of Technology, Coimbatore, Tamilnadu, India. G.S. Nandakumar, Associate Professor, Department of Computer Science and Engineering, Kumaraguru College of Technology, Coimbatore, Tamilnadu, India.

Published By: Retrieval Number: K103609811S19/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.K1036.09811S19 169 & Sciences Publication

Achromatic Colouring of the Central Graph of Some Specialgraphs

and . Let be the root  The number of edges in = vertex. Allocate the names to the remaining vertices of the inner wheel and to the vertices of the outer wheel. In for , let II. OBSERVATION & RESULTS be the new vertex on the edge and let be the vertex on the edge joining Let be the For a wind mill graph , the achromatic number of newly introduced vertex on the line joining for is Let be the newly introduced vertex on Structural properties of the central graph of Antiprism the edge joining let be the newly graph introduced vertex on the edge joining .  The number of vertices in = Let and be two  The number of edges in = . colour sets. Allocate colour to the root vertex For  The maximum degree in = allocate to and to For creating an achromatic colouring, consider the colouring method as  The number of vertices in = follows:  The number of edges in =  For allocate to the vertex  For allocate the colour to the Theorem: 3 vertex . For an anti prism graph , the achromatic number of  Allocate the colour to the vertices and is  Allocate the colour to the vertex Proof:  For allocate to the vertex Consider the two sets of vertices and  For allocate the colour to the . Allocate the names in the set to the vertex . vertices of the inner cycle in the clock wise direction and

 Allocate the colour to the vertices and . allocate the names in the set to the vertices of the outer

 Allocate the colour to the vertex cycle in the same direction. If a new colour is given to any vertex, the other  In for let be the new vertex on the colours will not be adjacent to it. Hence the maximum line connecting , . number of colours for an achromatic colouring is  Let be the vertex on the line connecting

and . Example  Let be the newly created vertex on the line connecting , .  Let be the vertex on the line connecting and .  For , let be the newly created vertex on the line connecting and .  For , let be the newly created vertex on the line connecting .  Let be the vertex on the line connecting

Let and be two sets of colours.

For allocate to and to To make this an achromatic colouring, consider the following pattern of colouring:

 For allocate to the vertex ,

to the vertex and to

 For allocate to the vertex

and to the vertex and allocate to

Structural properties of the central graph of windmill  For allocate the new colour to the graph vertex

 The number of vertices in =  For allocate the new colour to

the vertex and allocate the colour to the vertex  The number of edges in = . By this allotment, the maximum possible number of

 The maximum degree in = colours for an achromatic colouring is

 The number of vertices in =

Published By: Retrieval Number: K103609811S19/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.K1036.09811S19 170 & Sciences Publication

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-11S, September 2019

Example

III. REFERENCES 1. Irving. R.W and Manlove. D.F. The b-Chromatic number of a graph. Discrete Applied Mathematics 91(1-3) 127- 141, 1999. 2. Jonathan gross and Jay yellan.Handbook of graph theory CRC press, New York, 2004. 3. Marko Jakovac and Sandi Klavzar. The b-chromatic number of cubic Graphs. Graphs and Combinatorics, 26:107-118, 2010. 4. Rokad A.H. Product cordial labelling of Double wheel and Double Fan Related Graphs, Kragujevac Journal of Matematics, Volume 43(1) 7-13, 2019. 5. Thilagavathy. K.P and Santha. A. A comparative study on achromatic and b-chromatic number of certain graphs. Italian Journal of Pure and Applied Mathematics”, 39- 44, 2018. 6. Nagaraj Balakrishnan, Reshmi S., and R. Arunkumar. "Smart real time rescue system for fishermen." Pak J Biotechnol 15, no. 1 (2018): 73-75. 7. Thilagavathy. K.P and Santha. A. The Achromatic and b- Chromatic Colouring of Central Graph of Book Graph and Shadow graph of Path graph.International Journal of pure and applied mathematics, 113,1-9, 2017. 8. Vernold Vivin, J. Thilagavathi, K, ”On Harmonious colouring of Central Graphs”, Far East. Math. Sci (FJMS), 2, pp189-197, 2006.

Published By: Retrieval Number: K103609811S19/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijitee.K1036.09811S19 171 & Sciences Publication