International Journal of Mathematics and Mathematical Sciences

Volumeย 2011ย (2011), Article IDย 279246, 5 pages

http://dx.doi.org/10.1155/2011/279246

## Equitable Coloring on Total Graph of Bigraphs and Central Graph of Cycles and Paths

^{1}Department of Mathematics, University College of Engineering Nagercoil, Anna University of Technology Tirunelveli (Nagercoil Campus), Nagercoil 629 004, Tamil Nadu, India^{2}Department of Mathematics, R.V.S College of Engineering and Technology, Coimbatore 641 402, Tamil Nadu, India^{3}Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore 641 062, Tamil Nadu, India

Received 2 December 2010; Accepted 9 February 2011

Academic Editor: Marcoย Squassina

Copyright ยฉ 2011 J. Vernold Vivin et al. 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

The notion of equitable coloring was introduced by Meyer in 1973. In this paper we obtain interesting results regarding the equitable chromatic number for the total graph of complete bigraphs , the central graph of cycles and the central graph of paths .

#### 1. Introduction

The central graph [1, 2] of a graph is formed by adding an extra vertex on each edge of , and then joining each pair of vertices of the original graph which were previously nonadjacent.

The total graph [3, 4] of has vertex set and edges joining all elements of this vertex set which are adjacent or incident in .

If the set of vertices of a graph can be partitioned into classes such that each is an independent set and the condition holds for every pair (), then is said to be *equitably k-colorable*. The smallest integer for which is equitable -colorable is known as the *equitable chromatic number* [5โ10] of and denoted by . Additional graph theory terminology used in this paper can be found in [3, 4].

#### 2. Equitable Coloring on Total Graph of Complete Bigraphs

Theorem 2.1. *If , the equitable chromatic number of total graph of complete bigraphs ,
*

*Proof. *Let be the bipartition of , where and . Let be the edges of . By the definition of total graph, has the vertex set and the vertices induce disjoint cliques of order in . Also is adjacent to .*Case 1 (if , ). *Now we partition the vertex set as follows:
Clearly are independent sets and satisfying the condition , for any , . Since there exists a clique of order in . , also each of receives one color different from the color class assigned to the clique induced by . By the definition of total graph, each is adjacent with . Therefore, and are independent sets and hence . That is, ; therefore . Hence .*Case 2 (if , ). *Now we partition the vertex set as follows:
Clearly are independent sets of . Also and satisfy the condition , for any , . Since there exists a clique of order in . , that is, , therefore . Hence .

#### 3. Equitable Coloring on Central Graph of Cycles and Paths

Theorem 3.1. *If , the equitable chromatic number of central graph of cycles ,
*

*Proof. *Let and be the vertices and edges of taken in the cyclic order. By the definition of central graph, has the vertex set , where is the vertex of subdivision of the edge and joining all the nonadjacent vertices of in .*Case 1 ( is odd). *We partition the vertex set as
Clearly are independent sets of . Also and . The inequality holds, for any , . For each , is nonadjacent with and and hence . That is, , . Therefore, .*Case 2 ( is even). *Now we partition the vertex set as follows:
Clearly are independent sets of . Also . The inequality holds, for any , . For each , is nonadjacent with and and hence . That is, , . Therefore, .

*Remark 3.2. *If , then , respectively.

Theorem 3.3. *If , the equitable chromatic number of central graph of paths ,
*

*Proof. *Let and be the vertices and edges of . By the definition of central graph, has the vertex set , where is the vertex of subdivision of the edge and joining all nonadjacent vertices of in .*Case 1 ( is odd). *Now we partition the vertex set as follows:
Clearly are independent sets of . Also and . The inequality holds, for any , . For each , is nonadjacent with and and hence . That is, , . Therefore .*Case 2 ( is even). *Now we partition the vertex set as follows:
Clearly are independent sets of . Also and . The inequality holds for any , . For each , is nonadjacent with and and hence . That is, , . Therefore, .

*Remark 3.4. *If , then , respectively.

#### References

- A. M. M. Akbar, K. Kaliraj, and J. Vernold Vivin, โOn equitable coloring of central graphs and total graphs,โ in
*International Conference on Graph Theory and Its Applications*, vol. 33 of*Electronic Notes in Discrete Mathematics*, pp. 1โ6, Elsevier Science, Amsterdam, The Netherlands, 2009. View at Google Scholar - J. Vernold Vivin,
*Harmonious coloring of total graphs, n-leaf, central graphs and circumdetic graphs*, Ph.D. thesis, Bharathiar University, Coimbatore, India, 2007. - J. A. Bondy and U. S. R. Murty,
*Graph Theory with Applications*, MacMillan, London, UK, 1976. - F. Harary,
*Graph Theory*, Addison-Wesley, Reading, Mass, USA, 1969. - B.-L. Chen and K.-W. Lih, โEquitable coloring of trees,โ
*Journal of Combinatorial Theory. Series B*, vol. 61, no. 1, pp. 83โ87, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - H. Furmaczyk and M. Kubale, โThe complexity of equitable Vertex coloring of graphs,โ
*Journal of Applied Computer Science*, vol. 13, no. 2, pp. 95โ107, 2005. View at Google Scholar - H. L. Bodlaender and F. V. Fomin, โEquitable colorings of bounded treewidth graphs,โ
*Theoretical Computer Science*, vol. 349, no. 1, pp. 22โ30, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - K.-W. Lih and P.-L. Wu, โOn equitable coloring of bipartite graphs,โ
*Discrete Mathematics*, vol. 151, no. 1–3, pp. 155โ160, 1996. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - M. Kubale,
*Graph Colorings*, vol. 352 of*Contemporary Mathematics*, American Mathematical Society, Providence, RI, USA, 2004. - W. Meyer, โEquitable coloring,โ
*The American Mathematical Monthly*, vol. 80, pp. 920โ922, 1973. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH