A conjecture of Neumann-Lara on infinite families of rr-dichromatic circulant tournaments

In this paper, we exhibit infinite families of vertex critical rr-dichromatic circulant tournaments for all r≥3r≥3. The existence of these infinite families was conjectured by Neumann-Lara [V. Neumann-Lara, Note on vertex critical 4-dichromatic circulant tournaments, Discrete Math. 170 (1997) 289–291], who later proved it for all r≥3r≥3 and r≠7r≠7. Using different methods, we provide new constructions of such infinite families for all r≥3r≥3, which covers the case r=7r=7 and thus settles the conjecture.