31 research outputs found
On product Schur triples in the integers
Schur's theorem states that in any -colouring of the set of integers
there is a monochromatic solution to , provided is sufficiently
large. Abbott and Wang studied the size of the largest subset of such
that there is a -colouring avoiding a monochromatic . In other
directions, the minimum number of in -colourings of and the
probability threshold in random subsets of for the property of having a
monochromatic in any -colouring were investigated. In this paper, we
study natural generalisations of these streams to products , in a
deterministic, random, and randomly perturbed environments.Comment: 13 page
Cycle factors in randomly perturbed graphs
We study the problem of finding pairwise vertex-disjoint copies of the ω>-vertex cycle Cω>in the randomly perturbed graph model, which is the union of a deterministic n-vertex graph G and the binomial random graph G(n, p). For ω>≥ 3 we prove that asymptotically almost surely G U G(n, p) contains min{δ(G), min{δ(G), [n/l]} pairwise vertex-disjoint cycles Cω>, provided p ≥ C log n/n for C sufficiently large. Moreover, when δ(G) ≥ αn with 0 ≤ α/l and G and is not 'close' to the complete bipartite graph Kαn(1 - α)n, then p ≥ C/n suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that p ≥ C/n suffices when α > n/l for finding [n/l] cycles Cω>. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson-Kahn-Vu Theorem for Cω>-factors and the resolution of the El-Zahar Conjecture for Cω>-factors by Abbasi