Random perturbation of sparse graphs

Hahn‐Klimroth, Max; Maesaka, Giulia S.; Mogge, Yannick; Mohr, Samuel; Parczyk, Olaf

doi:10.48550/arxiv.2004.04672

Cited by 1 publication

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we prove Theorem 1.3. The main ingredients of our proof are Theorem 1.1, a randomness shift argument, taken from [31], which shifts randomness, so to speak, 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 [14]).…”

Section: Prescribed Rainbow Spanning Trees In Randomly Perturbed Graphsmentioning

confidence: 99%

“…The study of the emergence of various spanning configurations in randomly perturbed (hyper)graphs dates back to the work of Bohman, Frieze, and Martin [11], and received significant attention in recent years, see e.g. [8,9,10,13,14,18,27,28,30,31,32] 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 [16] and random regular graphs [17].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rainbow trees in uniformly edge-coloured graphs

Aigner-Horev,

Hefetz,

Lahiri

2021

Preprint

View full text Add to dashboard Cite

We obtain sufficient conditions for the emergence of spanning and almost-spanning boundeddegree rainbow trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of G(n, ω(1)/n), using a palette of size n, a.a.s. admits a rainbow copy of any given boundeddegree tree on at most (1 − ε)n vertices, where ε > 0 is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in G(n, C/n), where C > 0 is independent of n.Given an n-vertex graph G with minimum degree at least δn, where δ > 0 is fixed, we use our aforementioned result in order to prove that a uniform colouring of the randomly perturbed graph G ∪ G(n, ω(1)/n), using (1 + α)n colours, where α > 0 is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich, Kwan, and Sudakov who proved that G ∪ G(n, C/n), where C > 0 is independent of n, a.a.s. admits a copy of any given bounded-degree spanning tree.Finally, and with G as above, we prove that a uniform colouring of G ∪ G(n, ω(n −2 )) using n − 1 colours a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

show abstract

Section: Prescribed Rainbow Spanning Trees In Randomly Perturbed Graphsmentioning

confidence: 99%