The Bermond-Thomassen conjecture

Jean-Sébastien Sereni

Abstract

In 1981, Bermond and Thomassen conjectured that every digraph of minimum out-degree at least 2k-1 contains k vertex-disjoint directed cycles. This conjecture is a simple observation if k is 1, and was proved by Thomassen when k is 2 in 1983. I will survey the (few) results known about the conjecture, and sketch a proof that it holds when k is three. To this end, I will first show that Thomassen's result can be improved. This is joint work with Nicolas Lichiardopol and Attila Por.