297 research outputs found
Universality for bounded degree spanning trees in randomly perturbed graphs
We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph G α on n vertices with δ(G α ) ≥ αn for α > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph G α ∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ
Embedding spanning bounded degree graphs in randomly perturbed graphs
We study the model G 8 G(n; p) of randomly perturbed dense graphs, where G is any n-vertex graph with minimum degree at least n and G(n; p) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every > 0 and C 5, and every n-vertex graph F with maximum degree at most , we show that if p = !(n−2~(+1)) then G 8 G(n; p) with high probability contains a copy of F. The bound used for p here is lower by a log-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in G(n; p) alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the rst example of graphs where the appearance threshold in G 8 G(n; p) is lower than the appearance threshold in G(n; p) by substantially more than a log-factor. We prove that, for every k C 2 and > 0, there is some > 0 for which the kth power of a Hamilton cycle with high probability appears in G 8 G(n; p) when p = !(n−1~k−). The appearance threshold of the kth power of a Hamilton cycle in G(n; p) alone is known to be n−1~k, up to a log-term when k = 2, and exactly for k > 2
Random perturbation of sparse graphs
In the model of randomly perturbed graphs we consider the union of a deterministic graph Gα with minimum degree αn and the binomial random graph 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 G(n, p). In this note we extend this result in Gα ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. Gα ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part 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 G(n, p)
Rainbow subgraphs of uniformly coloured randomly perturbed graphs
For a given , the randomly perturbed graph model is defined
as the union of any -vertex graph with minimum degree and
the binomial random graph on the same vertex set. Moreover,
we say that a graph is uniformly coloured with colours in if each
edge is coloured independently and uniformly at random with a colour from
.
Based on a coupling idea of McDiarmird, we provide a general tool to tackle
problems concerning finding a rainbow copy of a graph in a uniformly
coloured perturbed -vertex graph with colours in . For
example, our machinery easily allows to recover a result of Aigner-Horev and
Hefetz concerning rainbow Hamilton cycles, and to improve a result of
Aigner-Horev, Hefetz and Lahiri concerning rainbow bounded-degree spanning
trees.
Furthermore, using different methods, we prove that for any and integer , there exists such that the
following holds. Let be a tree on vertices with maximum degree at most
and be an -vertex graph with . Then a
uniformly coloured with colours in
contains a rainbow copy of with high probability. This is optimal both in
terms of colours and edge probability (up to a constant factor).Comment: 22 pages, 1 figur
- …