Infinite families of -dichromatic vertex critical circulant tournaments

In this talk we expose the results about 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 find explicit constructions of these infinite families for all r ⩾3, including the case when r = 7 , which complete the proof of the conjecture.