Extremal problems of long cycles in random graphs

In this talk, we consider the random version of some classical extremal problems in the context of long cycles. This type of problems can also be seen as random analogues of the Turán number of long cycles, established by Woodall in 1972.

For a graph $G$ on $n$ vertices and a graph $H$, denote by $\text{ex}(G,H)$ the maximal number of edges in an $H$-free subgraph of $G$. We consider a random graph $G\sim G(n,p)$ where $p>C/n$, and determine the asymptotic value of $\text{ex}(G,C_t)$, for every $A\log(n)< t< (1- \varepsilon)n$. The behaviour of $\text{ex}(G,C_t)$ can depend substantially on the parity of $t$. In particular, our results match the classical result of Woodall, and demonstrate the transference principle in the context of long cycles.

Using similar techniques, we also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density (a nearly ''Woodall graph"). If time permits, we will present some connections to size-Ramsey numbers of long cycles.

Based on joint works with Michael Krivelevich and Adva Mond.