Acyclic sets and colorings in digraphs under restrictions on degrees and cycle lengths

Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which $V(D)$ can be partitioned. In this paper, we study $\vec\alpha(D)$ and $\vec\chi(D)$ from various perspectives, including restrictions on degrees and cycle lengths. A main result is that, if $D$ is a random $r$-regular digon-free simple digraph of order $n$, then $\vec{\alpha}(D) = \Theta(n \log r /r)$ with high probability. This corresponds to a result of Spencer and Subramanian on the Erd\H{o}s--R\'{e}nyi random digraph model. Along the way, we derive some related results and propose some conjectures. An example of this is an analogue of the theorem of Bondy which bounds the chromatic number of a graph by the circumference of any strong orientation.