Vertex-Disjoint Cycles of the Same Length

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 of the known result that if k> 2, there is an integer nk such that any graph of order at least nk and minimum degree at least 2k contains k vertex-disjoint cycles of the same length.