On oriented cycles in randomly perturbed digraphs

TL;DR

This paper extends the understanding of cycle orientations in randomly perturbed digraphs, showing that certain degree conditions almost surely guarantee the presence of all cycle orientations, with relaxed conditions for specific cycle types.

Contribution

It generalizes previous results by proving the existence of all cycle orientations under degree conditions and introduces a relaxation for certain cycle types.

Findings

Almost sure existence of all cycle orientations under degree conditions

Relaxed degree conditions for cycles with few vertices of indegree 1

Use of a variant of Montgomery's absorbing method

Abstract

In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha>0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $\alpha n$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$. Our proofs make use of a variant of an absorbing method of Montgomery.