For a digraph D, let c(D) be the largest integer k for which there are directed cycles C_1,…,C_k through a common vertex v such that C_1-v,…,C_k-v are pairwise vertex-disjoint. For r∈N, let c_r be the minimum of c(D) over all r-regular digraphs. Mader's conjecture. For every k∈N there exists some r_0∈N such that c_r≥ k for every r≥ r_0; equivalently, lim_r rightarrow∞c_r=∞. The paper proves this conjecture in the stronger form c_r≥ lceil frac322r rceil for every r∈N, so the conjecture is solved.

Mader's conjecture on openly disjoint cycles in regular digraphs

Mader conjectured in 2008 that the minimum number of openly disjoint directed cycles through a vertex in an $r$ -regular digraph tends to infinity as $r$ grows. Seymour posed the regular-digraph question in 2005.

Known results

Thomassen (1985) constructed digraphs with arbitrarily large minimum in- and out-degree but with $c(D)\le 2$ .
Mader (2008) proved the case $r=3$ .
Mader (2008) constructed examples with $c(D)\le r-1$ for every $r\ge 8$ .
The cases $r\in\{4,\ldots,7\}$ remain open for the stronger equality question $c_r=r$ .

September 2026 scan: claimed linear bound

Version $2$ of the preprint claims that every $r$ -regular digraph satisfies $c(D)\ge\lceil 3r/22\rceil$ , which proves Mader’s conjecture, and further claims $c_r\le 7\lceil r/8\rceil$ and $3/22\le\lim_{r\to\infty}c_r/r\le 7/8$ . Its posting date is not stated in the retrieved material, and the claim remains unverified.

Current status (as of September 2026): Mader’s conjecture is claimed solved by the version- $2$ preprint, but its proof has not been independently verified; the separate equality question $c_r=r$ for $r\in\{4,\ldots,7\}$ remains open.

Mader's conjecture on openly disjoint cycles in regular digraphs

Primary source

Progress summary

Solutions 0