A well known generalisation of Dirac's theorem states that if a graph G on $n\ge 4k$ vertices has minimum degree at least $n/2$ then G contains a 2-factor consisting of exactly k cycles. This is easily seen to be tight in terms of the bound on the minimum degree. However, if one assumes in addition that G is Hamiltonian it has been conjectured that the bound on the minimum degree may be relaxed. This was indeed shown to be true by Sárközy. In subsequent papers, the minimum degree bound has been improved, most recently to $(2/5+\epsilon )n$ by DeBiasio, Ferrara, and Morris. On the other hand no lower bounds close to this are known, and all papers on this topic ask whether the minimum degree needs to be linear. We answer this question, by showing that the required minimum degree for large Hamiltonian graphs to have a 2-factor consisting of a fixed number of cycles is sublinear in n.

狄拉克定理的一个众所周知的概括是，如果图G在$ñ\ge 4ķ$ 顶点至少具有最小度 $ñ/2$那么G包含2个因子，恰好由k个周期组成。从最小程度的界限来看，这很容易看出来。但是，如果另外假设G是哈密顿量，则可以推测最小度的界限可以放宽。Sárközy确实证明了这一点。在随后的论文中，最小度界已得到改进，最近是$（2/5+\epsilon ）ñ$由DeBiasio，Ferrara和Morris撰写。另一方面，没有下界是已知的，有关该主题的所有论文都询问最小度是否需要是线性的。通过显示大汉密尔顿图具有2个因子（由固定数量的循环组成）所需的最小度在n中为次线性，我们对此问题进行了回答。