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

In this paper, we exhibit infinite families of vertex critical r -dichromatic circulant tournaments for all r ≥ 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 ≥ 3 and r ≠ 7 . Using different methods, we provide new constructions of such infinite families for all r ≥ 3 , which covers the case r = 7 and thus settles the conjecture.