Large Hamiltonian graphs with minimum degree n to the power 1 minus a small epsilon contain a 2-factor consisting of exactly k cycles.
Chiba, On degree sum conditions for 2-factors with a prescribed number of cycles.Discrete Math.341.10 (2018)
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On $2$-factors of Hamiltonian graphs
Large Hamiltonian graphs with minimum degree n to the power 1 minus a small epsilon contain a 2-factor consisting of exactly k cycles.