Vertex-disjoint cycles of the same length

Corradi and Hajnal proved that a graph of order at least <f>3k</f> and minimum degree at least <f>2k</f> contains <f>k</f> vertex-disjoint cycles. Ha¨ggkvist 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 of the known result that if <f>k>2</f>, there is an integer <f>nk</f> such that any graph of order at least <f>nk</f> and minimum degree at least <f>2k</f> contains <f>k</f> vertex-disjoint cycles of the same length. [Copyright &y& Elsevier]