A step toward the Bermond-Thomassen conjecture about disjoint cycles in digraphs

Nicolas Lichiardopol (1) , Attila Pór (2) , Jean-Sébastien Sereni (3, 4)

1 Lycée Adam de Craponne
2 WKU - Department of Mathematics - Western Kentucky University
3 KAM - Department of Applied Mathematics (KAM)
4 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

In 1981, Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k-1 contains k disjoint cycles. This conjecture is trivial for k=1, and was established for k=2 by Thomassen in 1983. We verify it for the next case, proving that every digraph with minimum out-degree at least five contains three disjoint cycles. To show this, we improve Thomassen's result by proving that every digraph whose vertices have out-degree at least three, except at most two with out-degree two, indeed contains two disjoint cycles.

Mots clés: digraph; disjoint cycles; Bermond-Thomassen conjecture; out-degree

Domaines: Mathématique discrète [cs.DM]

Nicolas Lichiardopol, Attila Pór, Jean-Sébastien Sereni. A step toward the Bermond-Thomassen conjecture about disjoint cycles in digraphs. SIAM Journal on Discrete Mathematics, 2009, 23 (2), pp.979--992. ⟨10.1137/080715792⟩. ⟨hal-00487102⟩