Independence number and vertex-disjoint cycles

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, let f (k, ) be the maximum order of a graph G with independence number (G) , which has no k vertex-disjoint cycles. We prove that f (k, ) = 3k + 2 − 3 if 1 5 or 1 k 2, and f (k, ) 3k + 2 − 3 in general. We also prove the following results: (1) there exists a constant c (depending only on ) such that f (k, ) 3k + c , (2) there exists a constant tk (depending only on k) such that f (k, ) 2 + tk , and (3) there exists no absolute constant c such that f (k, ) c(k + ). © 2006 Published by Elsevier B.V.