Large feedback arc sets, high minimum degree subgraphs, and long cycles in Eulerian digraphs

TL;DR

This paper establishes tight bounds on feedback arc sets, cycle lengths, and minimum degree subgraphs in Eulerian digraphs, advancing understanding of their structural properties.

Contribution

It introduces new bounds for feedback arc sets, cycle lengths, and minimum degree subgraphs in Eulerian digraphs, and provides methods to find long cycles.

Findings

Lower bound for feedback arc set size: (m^2/2n^2 + m/2n)

Existence of short cycles: at most 6n^2/m length

Existence of high minimum degree Eulerian subgraphs

Abstract

A minimum feedback arc set of a directed graph $G$ is a smallest set of arcs whose removal makes $G$ acyclic. Its cardinality is denoted by $\beta(G)$. We show that an Eulerian digraph with $n$ vertices and $m$ arcs has $\beta(G) \ge m^2/2n^2+m/2n$, and this bound is optimal for infinitely many $m, n$. Using this result we prove that an Eulerian digraph contains a cycle of length at most $6n^2/m$, and has an Eulerian subgraph with minimum degree at least $m^2/24n^3$. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollob\'as and Scott, we also show how to find long cycles in Eulerian digraphs.