Openly disjoint cycles and directed tree-width of regular digraphs

Given a digraph $D$, let $c(D)$ denote the largest integer $k$ such that there are $k$ openly disjoint cycles through a vertex, i.e., a collection of directed cycles $C_1,\ldots,C_k$ through a common vertex $v$ such that $C_1-v,\ldots,C_k-v$ are pairwise vertex-disjoint. The famous Caccetta-H\"aggkvist conjecture and its regular variant due to Behzad, Chartrand and Wall from 1970, have motivated the study of degree conditions forcing $c(D)$ to be large. In 1985 Thomassen constructed digraphs of arbitrarily high minimum out- and in-degree such that $c(D)\le 2$. In 2005, Seymour asked whether in contrast every $r$-regular digraph satisfies $c(D)=r$, which would have implied the Behzad-Chartrand-Wall conjecture. In 2008, Mader answered this negatively for every $r\ge 8$, but conjectured that nevertheless the minimum value $c_r$ of $c(D)$ over all $r$-regular digraphs grows with $r$, i.e. $\lim_{r\rightarrow\infty}c_r=\infty$. As the first main result of our paper, we prove Mader's conjecture in a strong form by showing $c_r\ge \lceil\frac{3}{22} r\rceil$ for every $r\in \mathbb{N}$. We also show $c_r\le 7\left\lceil \frac{r}{8}\right\rceil$, improving the previous best upper bound $c_r\le r-\Theta(\sqrt{r})$ due to Mader. In our second main result we show that every $r$-regular digraph has directed tree-width $\Omega(r)$. This is tight up to the implied constant and cannot be extended to digraphs of minimum out- and in-degree at least $r$. As a corollary we obtain the existence of a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that every regular digraph with degree at least $f(k)$ contains a subdivision of the cylindrical wall of order $k$, and hence of a large class of planar digraphs. This makes progress on the notoriously difficult problem of finding degree conditions guaranteeing subdivisions of digraphs, related to a well-known conjecture of Mader from 1985.