Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

TL;DR

This paper proves a tight upper bound on the minimum feedback arc set size in oriented multigraphs with maximum degree 5, improving previous bounds and confirming a conjecture for such graphs.

Contribution

It establishes a new tight upper bound of one-third of the total arcs for feedback arc sets in bounded-degree multigraphs, strengthening Hanauer's conjecture.

Findings

Proves ${ m fas}(D) \,\le\, m/3$ for degree-5 multigraphs.

Improves previous bounds by Berger and Shor.

Identifies bounds for the constant c in ${ m fas}(D) \,\le\, c n$.

Abstract

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.