Counting orientations of random graphs with no directed k-cycles

TL;DR

This paper determines the approximate number of orientations of random graphs that avoid directed cycles of a fixed length, resolving a previously posed conjecture in graph theory.

Contribution

It establishes the growth rate of such orientations for all fixed cycle lengths, up to polylogarithmic factors, confirming a conjecture by Kohayakawa, Morris, and the authors.

Findings

Established the order of growth of orientations with no directed k-cycle

Resolved a conjecture in the field of random graph orientations

Provides bounds up to polylogarithmic factors

Abstract

For every $k \geq 3$, we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length $k$. This solves a conjecture of Kohayakawa, Morris and the last two authors.