AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 45 (2009), Pages 25–35 Enumeration and dichromatic number of tame tournaments V´ ıctor Neumann-Lara ∗ Mika Olsen † Instituto de Matem´ aticas Universidad Nacional Aut´ onoma de M´ exico M´ exico D. F. M´ exico Abstract The concept of molds, introduced by the authors in a recent preprint, break regular tournaments naturally into big classes: cyclic tournaments, tame tournaments and wild tournaments. We enumerate completely the tame molds, and prove that the dichromatic number of a tame tourna- ment is 3. 1 Introduction A mold is a regular tournament M such that all the paths of the domination digraph D (M) are of order at most 2. The molds were defined and studied in [12]. Two vertices u, v form a dominant pair of a regular tournament T if N - (u, T \{u, v})= N + (v,T \{u, v}) [12]. The domination graph of a tournament T , denoted by dom(T ), is the graph on the vertex set V (T ) with edges between dominant pairs of T . The domination graph was defined in [7] and the domination graph of regular tournaments were characterized by Cho et al. in [4, 5]. Recently the authors have given a simpler proof of this characterization by the use of molds [12]. The domination digraph of a tournament T , denoted by D (T ), is the domination graph on the vertex set V (T ) with the orientation induced by T . The domination digraph was defined in [6]. Since the paths of a domination graph dom(T ) are all directed in the tournament T , the characterization of domination graphs induces a characterization of domination digraphs. ∗ Passed away February 2004 † Current address: Departamento de Matem´ aticas Aplicadas y Sistemas, Universidad Aut´ onoma Metropolitana, Unidad Cuajimalpa, M´ exico D.F., M´ exico