Disjoint cycles in graphs with restricted independence number

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In 1963, Corrádi and Hajnal proved that every graph with at least 3 k vertices and minimum degree at least 2 k contains a collection of k vertex-disjoint cycles. The sharpness examples for this theorem were characterized by Kierstead, Kostochka, and Yeager in 2017 and one consequence of this characterization is that when k ≥ 3 , every graph with n ≥ 3 k vertices, minimum degree at least 2 k − 1 , and independence number at most n − 2 k − 1 has k vertex-disjoint cycles. We extend this result by showing that there exists β > 0 and t 0 such that for every t ≥ t 0 , k ≥ 25 t and n ≥ 4 k + t , every graph on n vertices with minimum degree at least 2 k − t and independence number at most n − 2 k − t + β t log t contains a collection of k vertex-disjoint cycles. We also show that the condition on the independence number is sharp up to the constant β .