Stability for Vertex Cycle Covers

Abstract

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.

Embedding graphs into larger graphs: results, methods, and problems

“…See e.g. Balogh, Mousset, Skokan, [80], D. Ellis, [256], E. Friedgut [362], Füredi, Kostochka and Luo [375], Füredi, Kostochka, Luo and Verstraëte [377,376] W.T. Gowers and O. Hatami [408], Gyárfás, G.N.…”

“…Burr and Erdős conjectured [164] that the 2-colour Ramsey number R(H n , H n ) is linear in n for bounded degree graphs. 80 First some weaker bound was found by József Beck, but then the conjecture was proved by Theorem 5.14 (Chvátal-Rödl-Szemerédi-Trotter (1983), [187]). For any ∆ > 0 there exists a constant Γ = Γ(∆) such that for any…”

“…The answer was given by 80 Here by linear we mean O(n). 81 Actually, Turán's corresponding results, or the Erdős-Sós type Ramsey-Turán theorems were not used in "applications", however the Ajtai-Komlós-Szemerédi type results are also in this category and, as we have seen in Sections 4.2-4.10, they were used in several beautiful and important results.…”

The square of a Hamilton cycle in randomly perturbed graphs

“…In fact, [7,Lemma 4.4] is weaker as, under the same assumptions, it gives a matching of size (α + 2d)t, but Lemma 3.6 follows from a straightforward adaption of its proof, which in turn builds on ideas from [4]. With Lemma 3.6, it is not hard to show that the reduced graph R can be vertex-partitioned into copies of stars K 1,k and matching edges K 1,1 , such that there are not too many stars.…”

Cycle factors in randomly perturbed graphs

“…Let G be a graph on n vertices with ℓ n and δ(G) ≥ (1 ℓ − γ)n that is not (1 ℓ, β)-stable. We apply the regularity lemma to G and obtain the reduced graph R, whose vertices are the clusters and there is an edge between two clusters if they give an (ε, d)-regular pair in G. By adapting ideas from [2] we proved the following stability result in [6].…”