LARGE RAINBOW CLIQUES IN RANDOMLY PERTURBED DENSE GRAPHS

Elad Aigner-Horev, Oran Danon, Dan Hefetz, Shoham Letzter

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)2975-2994
Number of pages20
JournalSIAM Journal on Discrete Mathematics
Volume36
Issue number4
DOIs
StatePublished - 2022

Keywords

  • anti-Ramsey
  • perturbed graphs
  • random graphs
  • thresholds

All Science Journal Classification (ASJC) codes

  • General Mathematics

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