TY - JOUR

T1 - Small rainbow cliques in randomly perturbed dense graphs

AU - Aigner-Horev, Elad

AU - Danon, Oran

AU - Hefetz, Dan

AU - Letzter, Shoham

N1 - Publisher Copyright:
© 2021 Elsevier Ltd

PY - 2022/3

Y1 - 2022/3

N2 - For two graphs G and H, write G⟶rbwH if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪G(n,p), where G is an n-vertex graph with edge-density at least d>0, and d is independent of n. In a companion paper, we proved that the threshold for the property G∪G(n,p)⟶rbwKℓ is n−1/m2(Kℓ/2), whenever ℓ≥9. For smaller ℓ, the thresholds behave more erratically, and for 4≤ℓ≤7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for ℓ∈{4,5,7} are n−5/4, n−1, and n−7/15, respectively. For ℓ∈{6,8} we determine the threshold up to a (1+o(1))-factor in the exponent: they are n−(2/3+o(1)) and n−(2/5+o(1)), respectively. For ℓ=3, the threshold is n−2; this follows from a more general result about odd cycles in our companion paper.

AB - For two graphs G and H, write G⟶rbwH if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪G(n,p), where G is an n-vertex graph with edge-density at least d>0, and d is independent of n. In a companion paper, we proved that the threshold for the property G∪G(n,p)⟶rbwKℓ is n−1/m2(Kℓ/2), whenever ℓ≥9. For smaller ℓ, the thresholds behave more erratically, and for 4≤ℓ≤7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for ℓ∈{4,5,7} are n−5/4, n−1, and n−7/15, respectively. For ℓ∈{6,8} we determine the threshold up to a (1+o(1))-factor in the exponent: they are n−(2/3+o(1)) and n−(2/5+o(1)), respectively. For ℓ=3, the threshold is n−2; this follows from a more general result about odd cycles in our companion paper.

UR - http://www.scopus.com/inward/record.url?scp=85114096400&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2021.103452

DO - 10.1016/j.ejc.2021.103452

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AN - SCOPUS:85114096400

SN - 0195-6698

VL - 101

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103452

ER -