On b-coloring of central graph of some graphs

Abstract

The b-chromatic number of G, denoted by ϕ(G), is the maximum k for which G has a b-coloring by k colors. A b-coloring of G by k colors is a proper k-coloring of the vertices of G such that in each color class i there exists a vertex x_{i} having neighbors in all the other k-1 color classes. Such a vertex x_{i} is called a b-dominating vertex, and the set of vertices {x₁,x₂…x_{k}} is called a b-dominating system. In this paper, we are going to investigate on the b-chromatic number of Central graph of Triangular Snake graph, Sunlet graph, Helm Graph, Double Triangular Snake graph, Gear graph, and Closed Helm graph are denoted as C(T_{n}), C(S_{n}), C(H_{n}), C(DT_{n}), C(G_{n}), C(CH_{n}) respectively.

Keywords

• b-coloring,b-chromatic number,central graph

References

1. Bondy, J.A. and Murty, U.S.R., Graph Theory with Applications. MacMillan, London, 1976.
2. Chartrand, G. and Zhang, P., Chromatic Graph Theory, Chapman and Hall, CRC, 2009.
3. Effantin, B., The b-chromatic number of power graphs of complete caterpillars, Journal of Discrete Mathematical Sciences and Cryptography, 8 (2005) 483-502.
4. Effantin, B. and Kheddouci, H., The b-chromatic number of some power graphs, Discrete Mathematics and Theoretical Computer Science, 6(2003) 45-54.
5. Gallian, A., A dynamic survey of Graph Labeling, 2009.
6. Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, (1999) 127-141, 91.
7. Jensen, T.R. and Toft, B., Graph Coloring Problems. Wiley-Interscience, 2009.
8. Somasundaram,S. Sandhya, S.S. and Viji, S.P., On Geometric Mean Graphs, International Mathematical Forum, 10(3), (2009) 115-125.

9. Vernold, Vivin J., Harmonious Coloring of Total graphs, n- leaf, Central graphs and circumdetic graphs, Ph.D Thesis, Bharathiyar University, Coimbatore, India, 2007.
10. Vernold, Vivin J., Venkatachalam, M. and Akbar, Ali M.M., A Note on Achromatic Coloring of Star Graph Families, Filomat, 23 (3)(2009) 251-255.
11. Vernold, Vivin J. and Venkatachalam, M., On b- chromatic number of Sunlet and Wheel Graph Families, Journal of the Egyptian Mathematical Society, 23(2) (2015) 215-218.
12. Yue, Xi, Yuansheng, Yang and Liping, Wang, On Harmonious Labeling of the Double Triangular Snake, Indian Journal of Pure and Applied Mathematics, (2008) 177-184, 39(2).