Abstract :
Let image be any graph, and also let image, image and image denote the maximum degree, the chromatic number and the independence number of image, respectively. A chromatic coloring of image is a proper coloring of image using image colors. A color class in a proper coloring of image is maximum if it has size image. In this paper, we prove that if a graph image (not necessarily connected) satisfies image, then there exists a chromatic coloring of image in which some color class is maximum. This cannot be guaranteed if image. We shall also give some other extensions.
