b-Chromatic number of some wheel related graphs

S. K. Vaidyaa,*, M. S. Shuklab

b-Chromatic number of some wheel related graphs

Authors :

S. K. Vaidya^a,*, M. S. Shukla^b

Author Address :

^aDepartment of Mathematics, Saurashtra University, Rajkot - 360005, Gujarat, India.
^bDepartment of Mathematics, Atmiya Institute of Technology and Science, Rajkot - 360005, Gujarat, India.

*Corresponding author.

Abstract :

A proper coloring $f$ is a $b$-coloring of the vertices of graph $G$ such that in each color class there exists a vertex that has neighbours in every other color classes. The $b$-chromatic number $varphi left( G ight)$ of a graph $G$ is the largest integer $k$ for which $G$ admits a $b$-coloring with $k$ colors. If $chi left( G ight)$ is the chromatic number of $G$ and $b$-coloring exists for every integer $k$ satisfying the inequality $chi left( G ight) le k le varphi left( G ight)$ then $G$ is called $b$-continuous. The $b$-spectrum $S_{b}(G)$ of a graph $G$ is the set of $k$ integers(colors) for which $G$ has a $b$-coloring. We investigate $b$-chromatic number for the graphs obtained from wheel $W_{n}$ by means of duplication of vertices. We also discuss $b$-continuity and $b$-spectrum for such graphs.

Keywords :

$b$-Coloring, $b$-Continuity, $b$-Spectrum.

DOI :

Article Info :

Received : May 15, 2014; Accepted : July 31, 2014.