1 DUCHET (Pierre) , JANAQI (Stefan), LESCURE (Françoise]., MAAMOUN (Molaz)., MEYNIEL (Henry), On contractions of digraphs onto K 3 *, Mat. Bohemica , to appear [2002-]. CONTRACTION OF DIGRAPHS ONTO K 3 ∗ . S. Janaqi LSD 2-IMAG, CNRS, Université Joseph Fourier, Grenoble, France. P. Duchet, F. Lescure, M. Maamoun and H. Meyniel Equipe Combinatoire, CNRS, Université Paris VI, Paris, France. Abstract : Extending the notion of a "minor" to directed graphs, we prove that any digraph with vertices, n 3, and m arcs, m≥ 3n-3, has K 3 ∗ as a minor. §1. Introduction. We extend the notion of a "minor" to directed graphs. In the non-oriented case the notion of contraction is well known (see [6] for instance). In this paper Hadwiger gave the following conjecture : If χ (G) = p then G is contractible onto K p . Dirac [3] proved this conjecture for p ≤ 4 . Wagner [10] showed that the case p=5 would be a consequence of the 4-color theorem ; Robertson, Seymour and Thomas [8] proved the case p=6. For a study of relationships between the existence of some minor of G and a generalisation of the notion of colouring of digraphs, the interested reader is referred to [7], where a directed version of Hadwiger’s conjecture is proposed. A sufficient condition for contractibility onto K 3 ∗ was given by Duchet and Kaneti [4] in terms of minimum in- and out- degrees. We prove here the following :