Cycle factors in randomly perturbed graphs

Böttcher, Julia, Parczyk, Olaf, Sgueglia, Amedeo and Skokan, Jozef (2022) Cycle factors in randomly perturbed graphs. Procedia Computer Science, 195. 404 - 411. ISSN 1877-0509

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Abstract

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.

Item Type:	Article
Official URL:	https://www.sciencedirect.com/journal/procedia-com...
Additional Information:	© 2021 Elsevier B.V.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	16 Feb 2022 15:45
Last Modified:	20 Sep 2025 02:08
URI:	http://eprintstest.lse.ac.uk/id/eprint/113760

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