Disproof of a Conjecture of Neumann-Lara

We disprove the following conjecture due to Vı́ctor Neumann-Lara: for every pair (r, s) of integers such that r > s > 2, there is an infinite set of circulant tournaments T such that the dichromatic number and the cyclic triangle free disconnection of T are equal to r and s, respectively. Let Fr,s denote the set of circulant tournaments T with dc(T ) = r and − →ω 3 (T ) = s. We show that for every integer s > 4 there exists a lower bound b(s) for the dichromatic number r such that Fr,s = ∅ for every r < b(s). We construct an infinite set of circulant tournaments T such that dc(T ) = b(s) and − →ω 3(T ) = s and give an upper bound B(s) for the dichromatic number r such that for every r > B(s) there exists an infinite set Fr,s of circulant tournaments. Some infinite sets Fr,s of circulant tournaments are given for b(s) < r < B(s).