TY - CHAP

T1 - Rainbow Cliques in Randomly Perturbed Dense Graphs

AU - Aigner-Horev, Elad

AU - Danon, Oran

AU - Hefetz, Dan

AU - Letzter, Shoham

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - For two graphs G and H, write G⟶ rbw H 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, and d is a constant that does not depend on n. We determine the threshold for the property G∪ G(n, p) ⟶ rbw Ks for every s. We show that for s≥ 9 the threshold is n-1/m2(K⌈s/2⌉) ; in fact, our 1-statement is a supersaturation result. This turns out to (almost) be the threshold for s= 8 as well, but for every 4 ≤ s≤ 7, the threshold is lower and is different for each 4 ≤ s≤ 7. Moreover, we prove that for every ℓ≥ 2 the threshold for the property G∪ G(n, p) ⟶ rbw C2 ℓ - 1 is n- 2 ; in particular, the threshold does not depend on the length of the cycle C2 ℓ - 1. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

AB - For two graphs G and H, write G⟶ rbw H 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, and d is a constant that does not depend on n. We determine the threshold for the property G∪ G(n, p) ⟶ rbw Ks for every s. We show that for s≥ 9 the threshold is n-1/m2(K⌈s/2⌉) ; in fact, our 1-statement is a supersaturation result. This turns out to (almost) be the threshold for s= 8 as well, but for every 4 ≤ s≤ 7, the threshold is lower and is different for each 4 ≤ s≤ 7. Moreover, we prove that for every ℓ≥ 2 the threshold for the property G∪ G(n, p) ⟶ rbw C2 ℓ - 1 is n- 2 ; in particular, the threshold does not depend on the length of the cycle C2 ℓ - 1. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

KW - Anti-Ramsey

KW - Random graphs

KW - Randomly perturbed graphs

UR - http://www.scopus.com/inward/record.url?scp=85114111937&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_25

DO - 10.1007/978-3-030-83823-2_25

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AN - SCOPUS:85114111937

T3 - Trends in Mathematics

SP - 154

EP - 159

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -