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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… (More)

Corradi and Hajnal proved that a graph of order at least 3k and minimum degree at least 2k contains k vertex-disjoint cycles. Häggkvist subsequently conjectured that a sufficiently large graph of minimum degree at least four contains two vertex-disjoint cycles of the same length. We prove that this conjecture is correct. In doing so, we give a short proof… (More)