About b-Coloring of Windmill Graph

Abstract

The b-chromatic number of a graph G is defined as the maximum number k of colors in a proper coloring to the vertices of G in such a way that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we show that for any windmill graph \(D_{n}^{m} ,\chi (M(D_{n}^{m} )) = \varphi (M(D_{n}^{m} ))\) and \(\chi (T(D_{n}^{m} )) = \varphi (T(D_{n}^{m} ))\).