Stability for vertex cycle covers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac’s Theorem: for any integer k ≥ 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\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 k1,…,k_r summing to k, the disjoint union I_k1n/k,k1 + · ·· + I_{kr n/k,kr} 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_k1n/k,k1 + · ·· + I_{kr n/k,kr}, for some sequence k₁,…,k_r of positive integers that sum to k. Our proof uses Szemerédi’s Regularity Lemma and the related machinery.