Long cycles in subgraphs of (pseudo)random directed graphs

TL;DR

This paper investigates the presence of long directed cycles in subgraphs of (pseudo)random directed graphs, establishing thresholds for cycle lengths based on edge density and demonstrating resilience properties.

Contribution

It introduces new bounds on the existence of long directed cycles in subgraphs of (pseudo)random directed graphs, highlighting their resilience to edge removal.

Findings

Subgraphs with slightly more than half the edges contain long cycles.

Existence of subgraphs with similar edge counts that lack long cycles.

Quantitative bounds on cycle lengths relative to total vertices.

Abstract

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every $0 < \gamma < 1/2$ we find a constant $c=c(\gamma)$ such that the following holds. Let $G=(V,E)$ be a (pseudo)random directed graph on $n$ vertices, and let $G'$ be a subgraph of $G$ with $(1/2+\gamma)|E|$ edges. Then $G'$ contains a directed cycle of length at least $(c-o(1))n$. Moreover, there is a subgraph $G''$ of $G$ with $(1/2+\gamma-o(1))|E|$ edges that does not contain a cycle of length at least $cn$.