Rainbow Cliques in Randomly Perturbed Dense Graphs

Abstract

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/m 2 (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 ≥ 2 the threshold for the property G∪G(n, p) rbw −→ C 2 −1 is n −2 ; in particular, the threshold does not depend on the length of the cycle C 2 −1 . It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

Small rainbow cliques in randomly perturbed dense graphs

“…The proof of Theorem 1.1 in [1] relies heavily on the so-called K LR-theorem [11,Theorem 1.6(i)]; the proofs of all the results stated above, employ entirely different approaches. Indeed, more refinement and control are required in order to handle small cliques.…”