Magnant–Martin path-cover conjecture for regular digraphs and oriented graphs

For a digraph $G$, let $\pi(G)$ denote the minimum number of vertex-disjoint directed paths needed to cover $V(G)$. A $d$-regular digraph has indegree and outdegree $d$ at every vertex; an oriented graph is a digraph with at most one directed edge between each pair of vertices.

Magnant–Martin's strengthened conjecture. If $G$ is a $d$-regular digraph on $n$ vertices, then

$$\pi(G)\leq \frac{n}{d+1}.$$

Moreover, if $G$ is oriented, then

$$\pi(G)\leq \frac{n}{2d+1}.$$

The paper proves these bounds in the dense setting, namely for sufficiently large $n$ when $d\geq\alpha n$ for fixed $f70b>0$. The conjecture is proposed as a stronger directed and oriented analogue of the regular-graph path-cover conjecture.