Tur\'an-type problems for long cycles in random and pseudo-random graphs

TL;DR

This paper investigates the maximum edges in subgraphs of random and pseudo-random graphs that avoid long cycles, extending classical Turán results to probabilistic settings and revealing parity-dependent behaviors.

Contribution

It determines the asymptotic Turán number for long cycles in random graphs and extends the results to pseudo-random graphs, demonstrating the transference principle in sparse settings.

Findings

Asymptotic value of ex(G(n,p), C_t) for p ≥ C/n and A log n ≤ t ≤ (1 - ε)n

Results depend on the parity of cycle length t

Extension of classical Turán results to random and pseudo-random graphs

Abstract

We study the Tur\'an number of long cycles in random graphs and in pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the asymptotic value of $ex(G(n,p), C_t)$ where $C_t$ is a cycle of length $t$, for $p\geq \frac Cn$ and $A \log n \leq t \leq (1 - \varepsilon)n$. The typical behavior of $ex(G(n,p), C_t)$ depends substantially on the parity of $t$. In particular, our results match the classical result of Woodall on the Tur\'an number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well. In fact, our techniques apply in a more general sparse pseudo-random setting. 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.