A note on vertex-disjoint cycles

Haggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k(2) + 19k + 10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k(2) cannot be decreased and, in this sense, this result is best possible.