In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.
Proof of the $(n/2-n/2-n/2)$ Conjecture for Large $n$