Vizing Bound for the Chromatic Number on Some Graph Classes

We are interested in hereditary classes of graphs such that every graph satisfies , where () denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (, diamond)-free, then , and we give examples to show that the bound is sharp.