Hall ratio of the Mycielski graphs

Let n(G) denote the number of vertices of a graph G and let (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by (G)=max { n(H) (H) : H ⊆ G } , where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies (G) (G) (G) where and denote the clique number and the chromatic number of G, respectively. We show that the interval [ (G), (G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs. © 2006 Elsevier B.V. All rights reserved.