Faudree–Gould–Jacobson–Lesniak–Saito conjecture on 2-factors in Hamiltonian graphs

Let $k\in\mathbb{N}$, and let $G$ be a Hamiltonian graph, meaning a graph containing a Hamilton cycle. A $2$-factor of $G$ is a spanning $2$-regular subgraph, equivalently a union of vertex-disjoint cycles containing every vertex of $G$.

Faudree–Gould–Jacobson–Lesniak–Saito conjecture. For any $k\in\mathbb{N}$ there are constants $c_k<1/2$, $n_k$ and $a_k$ such that any Hamiltonian graph $G$ of order $n\ge n_k$ with

$$\delta(G)\ge c_k n+a_k$$

contains a $2$-factor consisting of $k$ cycles.

The conjecture asks whether Hamiltonicity permits a minimum-degree threshold below Dirac's bound of $n/2$ while still guaranteeing a $2$-factor with exactly $k$ cycles. The surrounding discussion presents this as a conjecture of Faudree, Gould, Jacobson, Lesniak and Saito; no resolution is supplied here.