TY - JOUR

T1 - LARGE RAINBOW CLIQUES IN RANDOMLY PERTURBED DENSE GRAPHS

AU - Aigner-Horev, Elad

AU - Danon, Oran

AU - Hefetz, Dan

AU - Letzter, Shoham

N1 - Publisher Copyright:
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

PY - 2022

Y1 - 2022

N2 - For two graphs G and H, write (Equation presented). H if G has the property that every proper coloring 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. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property (Equation presented). In this paper, we show that for s ≥ 9 the threshold is (Equation presented). 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; see our companion paper for more details. Also in this paper, we determine that the threshold for the property (Equation presented). for every l ≤ 2; in particular, the threshold does not depend on the length of the cycle C2l - 1. For even cycles, and in fact any fixed bipartite graph, no random edges are needed at all; that is, (Equation presented). always holds, whenever G is as above and H is bipartite.

AB - For two graphs G and H, write (Equation presented). H if G has the property that every proper coloring 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. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property (Equation presented). In this paper, we show that for s ≥ 9 the threshold is (Equation presented). 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; see our companion paper for more details. Also in this paper, we determine that the threshold for the property (Equation presented). for every l ≤ 2; in particular, the threshold does not depend on the length of the cycle C2l - 1. For even cycles, and in fact any fixed bipartite graph, no random edges are needed at all; that is, (Equation presented). always holds, whenever G is as above and H is bipartite.

KW - anti-Ramsey

KW - perturbed graphs

KW - random graphs

KW - thresholds

UR - http://www.scopus.com/inward/record.url?scp=85148007427&partnerID=8YFLogxK

U2 - 10.1137/21M1423117

DO - 10.1137/21M1423117

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AN - SCOPUS:85148007427

SN - 0895-4801

VL - 36

SP - 2975

EP - 2994

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 4

ER -