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

TL;DR

This paper investigates the maximum size of acyclic vertex sets and their partitions in directed graphs, especially under degree and cycle length restrictions, providing probabilistic bounds and conjectures.

Contribution

It introduces new bounds on acyclic sets and colorings in digraphs with degree and cycle restrictions, extending existing probabilistic results.

Findings

For random r-regular digon-free digraphs, the maximum acyclic set size is Θ(n log r / r).

Derived related results and proposed conjectures on digraph colorings.

Extended Bondy's theorem to strong orientations with cycle length considerations.

Abstract

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.