On a Conjecture of Vيctor Neumann-Lara

We disprove the following conjecture due to Víctor Neumann-Lara: for every couple of integers ( r , s ) such that r ≥ s ≥ 2 there is an infinite set of circulant tournaments T such that the dichromatic number and the acyclic disconnection of T are equal to r and s respectively. We show that for every integer s ≥ 2 there exists a sharp lower bound b ( s ) for the dichromatic number r such that for every r < b ( s ) there is no circulant tournament T satisfying the conjecture with these parameters. We give an upper bound B ( s ) for the dichromatic number r such that for every r ≥ B ( s ) there exists an infinite set of circulant tournaments for which the conjecture is valid.