Random Perturbation of Sparse Graphs

Abstract

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

Hamiltonicity of graphs perturbed by a random geometric graph

“…A natural question arises from Theorem 1: What can we say about Hamiltonicity of randomly perturbed graphs when $$\alpha $$ is allowed to be a function of $$n$$ (which tends to 0)? This same question was recently considered by Hahn‐Klimroth, Maesaka, Mogge, Mohr and Parczy [14] when the random graph $$G$$ is binomial, and they extended the result of Bohman, Frieze and Martin [4] by proving that $$G=G(n,{\rm{\Theta }}(-\mathrm{log} \alpha \unicode{x02215}n))$$ is both sufficient and necessary. In view of Lemma 3, the following seems a natural conjecture (and, if true, it would be best possible up to the value of the constant $$C$$).…”

Rainbow trees in uniformly edge‐colored graphs

“…In this section we prove Theorem 1.3. The main ingredients of our proof are Theorem 1.1, a randomness shift argument (see details below), taken from [31], which allows one to apply the randomness originating from the random perturbation to the seed, and an absorbing structure; the latter can be viewed as a rainbow variant of the one used in [27] (see also [15]). Absorbers.…”

“…The study of the emergence of various spanning configurations in randomly perturbed (hyper)graphs dates back to the work of Bohman et al [12], and received significant attention in recent years, see for example, [5,[9][10][11]14,15,17,27,28,[30][31][32] to name just a few. In most results in this area, the random perturbation is the binomial random graph G(n, p) (or the, essentially equivalent, Erdős-Rényi random graph G(n, m)); however, quite recently other distributions have been considered, such as random geometric graphs [19] and random regular graphs [16].…”

Schur properties of randomly perturbed sets

“…One can also explore the behaviour of the perturbed model when the base set A is sparse. This direction has recently been explored in the graph setting [15] and aims to elucidate the full picture of how the randomly perturbed model transitions between the probabilistic and the extremal thresholds. In our setting, the result of Graham, Rödl and Ruciński (Theorem 1.2) determines the threshold if we have no deterministic elements while Theorem 1.3 gives that we can save some randomness when starting from a base set of size Ω(n).…”