Directed graphs without short cycles

TL;DR

This paper establishes bounds on feedback arc sets in directed graphs with large girth, extending previous results and answering questions about cycle lengths in such graphs.

Contribution

It proves a tight bound on feedback arc sets in high girth digraphs and explores cycle length properties related to feedback arc set size.

Findings

Bound on feedback arc set size in high girth digraphs.

Existence of cycles of almost any length in graphs with large feedback arc sets.

Structural characterization of graphs with large feedback arc sets.

Abstract

For a directed graph $G$ without loops or parallel edges, let $\beta(G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \subset E(G)$ such that $G \sm X$ has no directed cycles. Let $\gamma(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r \ge 4$ satisfies $\beta(G) \le c\gamma(G)/r^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed $0 < \theta < 1/2$ and sufficiently large $n$, if $G$ is a digraph with $n$ vertices and $\beta(G) \ge \theta n^2$, then for any $0 \le m \le \theta n-o(n)$ it contains a directed cycle whose length is between $m$ and $m+6 \theta^{-1/2}$. Moreover, there is a constant $C$ such that either $G$ contains directed cycles of every length between $C$ and $\theta n-o(n)$ or it is close to a digraph $G'$ with a simple structure: every strong component of $G'$ is periodic. These results are also tight up to the constant factors.