Disproof of a conjecture of Neumann-Lara

Abstract

We disprove the following conjecture due to Victor Neumann-Lara: for every pair (r; s) of integers such that r > s > 2, there is an in nite 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 w 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 in nite set of circulant tournaments T such that dc(T) = b(s) and w 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 in nite set Fr;s of circulant tournaments. Some in nite sets Fr;s of circulant tournaments are given for b(s) < r < B(s).