Bang-Jensen–Bessy–Thomassé conjecture on cycles and girth

The conjecture asserts that the minimum outdegree forcing $k$ vertex-disjoint directed cycles in a digraph of girth $g$ is $\left\lceil\frac{g}{g-1}k\right\rceil$.

May 2018 counterexamples

Yandong Bai and Yannis Manoussakis state in arXiv version 2, dated May 30, 2018, that they disprove the conjecture. For $g\geq 3$ and $t\geq 1$, they construct examples with minimum outdegree at least $\left\lceil\frac{g}{g-1}k\right\rceil$ but at most $k-t$ vertex-disjoint cycles, under stated lower bounds on $k$. They also obtain $f(k,4)\geq 2k-1$. The paper further rules out any fixed additive correction $c$ to the conjectured formula for all relevant $k$.

Current status (as of September 2026): The conjecture is claimed refuted by Bai and Manoussakis' counterexample construction, but the retrieved record provides no independent verification or later correction.