# 2-factors with k cycles in Hamiltonian graphs

## TL;DR

This paper proves that large Hamiltonian graphs can contain a 2-factor with a fixed number of cycles under a sublinear minimum degree condition, improving understanding of cycle structures in such graphs.

## Contribution

It establishes that the minimum degree condition for Hamiltonian graphs to contain a 2-factor with a fixed number of cycles is sublinear, advancing previous linear bounds.

## Key findings

- Minimum degree bound is sublinear for large Hamiltonian graphs.
- Existence of 2-factors with exactly k cycles under relaxed degree conditions.
- Improves upon previous linear minimum degree bounds.

## Abstract

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 S\'ark\"ozy. In subsequent papers, the minimum degree bound has been improved, most recently to $(2/5+\varepsilon)n$ 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 $n.$