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arXiv · 1202.2602

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

Hao Huang·Jie Ma·Asaf Shapira·Benny Sudakov·Raphael Yuster
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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.

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Hao Huang, Jie Ma, Asaf Shapira, Benny Sudakov, Raphael Yuster. 2012-02-13. Large feedback arc sets, high minimum degree subgraphs, and long cycles in Eulerian digraphs. https://arxiv.org/abs/1202.2602

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