Triangles in randomly perturbed graphs

Abstract

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any n-vertex graph G with linear minimum degree and the binomial random graph G(n, p). We prove that asymptotically almost surely G ∪ G(n, p) contains min{δ(G), ⌊n 3⌋} pairwise vertex-disjoint triangles, provided p ≥ C log n n, where C is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, to appear] in the case of triangle-factors. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159-176] this fully resolves the existence of triangle-factors in randomly perturbed graphs. We also prove a stability version of our result. Finally, we discuss further generalisations to larger clique-factors, larger cycle-factors, and 2-universality.

The square of a Hamilton cycle in randomly perturbed graphs

“…For α = 0, this is due to Ferber, Kronenberg, and Luh [14], for α ≥ 2 3 to Aigner and Brandt [1], and for α ∈ (0, 1 3 ) to [28]. The result in the range α ∈ [ 1 3 , 2 3 ) significantly strengthens one of our results from [7], where the same result was established for the containment of a triangle factor only.…”

“…The case k = 2 and t = 1 was already covered in [7,Theorem 2.4]. The general proof is similar and is given in Appendix A.…”

“…Assume that G is not ( 1 k+1 , β)-stable and let p ≥ Cn −(k−1) (2k−3) . Then we apply the regularity lemma to G and we use the following lemma on its reduced graph R. Lemma 3.6 (Lemma 4.4 in [7]). For any integer k ≥ 2, and 0 < β < 1 12 there exists d > 0 such that the following holds for any 0 < ε < d 4, 4β ≤ α ≤ 1 3 , and t ≥ 10 d .…”

“…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.…”

“…For certain values of α and r ≥ 3, the perturbed threshold is not yet precisely known for K r+1 -factors (see, e.g., [7] for more details). This suggests that determining the behaviour of the perturbed threshold in the entire range of α for the r-th power of a Hamilton cycle for r > 2 may be challenging.…”