Independence number and vertex-disjoint cycles

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