8 research outputs found
The anti-Ramsey threshold of complete graphs
For graphs and , let G {\displaystyle\smash{\begin{subarray}{c}
\hbox{\tiny\rm rb} \\ \longrightarrow \\ \hbox{\tiny\rm p}
\end{subarray}}}H denote the property that for every proper edge-colouring of
there is a rainbow in . It is known that, for every graph , an
asymptotic upper bound for the threshold function of this property for the random graph is
, where denotes the so-called maximum
-density of . Extending a result of Nenadov, Person, \v{S}kori\'c, and
Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower
bound for for . Furthermore, we show that .Comment: 19 page
Large rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every {\sl proper} colouring of its edges
yields a {\sl rainbow} copy of .
We study the thresholds for such so-called {\sl anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is a constant that does not depend on .
Our results in this paper, combined with our results in a companion paper,
determine the threshold for the property for every . In this paper, we
show that for the threshold is ; in fact, our -statement is a supersaturation result. This
turns out to (almost) be the threshold for as well, but for every , the threshold is lower; see our companion paper for more details.
In this paper, we also consider the property , and show that the
threshold for this property is for every ; in particular,
it does not depend on the length of the cycle . It is worth
mentioning that for even cycles, or more generally for any fixed bipartite
graph, no random edges are needed at all.Comment: 21 pages; some typos fixed in the last versio
Small rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every \emph{proper} colouring of its edges
yields a \emph{rainbow} copy of .
We study the thresholds for such so-called \emph{anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is independent of .
In a companion article, we proved that the threshold for the property
is
, whenever . For smaller , the thresholds behave more erratically, and for
they deviate downwards significantly from the aforementioned
aesthetic form capturing the thresholds for \emph{large} cliques.
In particular, we show that the thresholds for are
, , and , respectively. For we
determine the threshold up to a -factor in the exponent: they are
and , respectively. For , the
threshold is ; this follows from a more general result about odd cycles
in our companion paper.Comment: 37 pages, several figures; update following reviewer(s) comment