\(b\)-Chromatic Number of Triple Star Graph Families

D. Vijayalakshmi, M. Kalpana


A $b$-coloring of a graph \(G\) is a proper coloring of the vertices of \(G\) such that there exists a vertex in each color class joined to atleast a vertex in each other color class, such a vertex is called a dominating vertex. The \(b\)-chromatic number of a graph \(G\), denoted by \(b(G)\), is the maximal integer \(k\) such that \(G\) may have a \(b\)-coloring by \(k\) colors. In this paper, we investigate the \(b\)-chromatic number of Central graph, Middle graph, Total graph and Line graph of Triple Star graph, denoted by \(C(K_{1,n,n,n})\), \(M(K_{1,n,n,n})\), \(T(K_{1,n,n,n})\) and \(L(K_{1,n,n,n})\), respectively.


Central graph; Middle graph; Total graph; Line graph; Star graph; b-coloring; bchromatic number

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J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, MacMillan, London (1976).

B. Effantin, The b-chromatic number of power graphs of complete caterpillars, J. Discrete Math. Sci. Cryptogr. 8 (2005), 483 – 502.

B. Effantin and H. Kheddouci, The b-chromatic number of some power graphs, Discrete Math. Theor. Comput. Sci. 6 (2003), 45 – 54.

B. Effantin and H. Kheddouci, Exact values for the b-chromatic number of a power complete k-ary tree, J. Discrete Math. Sci. Cryptogr. 8 (2005), 117 – 129.

R.W. Irving and D.F. Manlove, The b-chromatic number of a graph, Discrete Appl. Math. 91 (1999), 127 – 141.

M. Kouider and A. El Sahili, About b-colouring of regular graphs, Rapport de Recherche No. 1432, CNRS-Universit´e Paris Sud LRI, 02/(2006).

S. Kla˘vzar and M. Jakovac, The b-chromatic number of cubic graphs, Preprint series 47 (January 27 2009), 1067.

V.J. Vernold, M. Venkatachalam and M.M. Akbar Ali, A note on achromatic coloring of star graph families, Filomat 23(3) (2009), 251 – 255.

V.J. Vernold and M. Venkatachalam, The b-chromatic number of star graph families, Le Mathematice (2010), 119 – 125.

DOI: http://dx.doi.org/10.26713%2Fjims.v9i3.1014

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