Cyclic Neighborhoods in Digraphs

Thilo Hartel; Dieter Rautenbach

Cyclic Neighborhoods in Digraphs

Combinatorics 2026-07-29 v1

Authors: Thilo Hartel , Dieter Rautenbach

Abstract

Let $D$ be a digraph of order $n$ and size $m$ with the property that no out-neighborhood or in-neighborhood of any vertex is acyclic. We show that, if $D$ is strongly connected, then $m\geq 7n/3$ , and, if $D$ is strongly $2$ -connected, then $m\geq 8n/3$ . Both results are best-possible.

Cite

@article{arxiv.2607.26606,
  title  = {Cyclic Neighborhoods in Digraphs},
  author = {Thilo Hartel and Dieter Rautenbach},
  journal= {arXiv preprint arXiv:2607.26606},
  year   = {2026}
}