Covering the edges of digraphs in $\mathscr{D}(3,3)$ and $\mathscr{D}(4,4)$ with directed cuts

TL;DR

This paper proves that the edges of any digraph in classes  and  can be covered by at most five directed cuts, and provides an example showing this bound is tight.

Contribution

It establishes the optimal bound for covering edges of certain digraph classes with directed cuts, advancing understanding of digraph edge coverings.

Findings

Edges of  and  digraphs can be covered with at most five directed cuts.

An example in  shows the bound is tight and cannot be improved.

The result improves bounds on edge coverings in specific digraph families.

Abstract

For nonnegative integers $k$ and $l$, let $\mathscr{D}(k,l)$ denote the family of digraphs in which every vertex has either indegree at most $k$ or outdegree at most $l$. In this paper we prove that the edges of every digraph in $\mathscr{D}(3,3)$ and $\mathscr{D}(4,4)$ can be covered by at most five directed cuts and present an example in $\mathscr{D}(3,3)$ showing that this result is best possible.