The acyclic and C⃗3-free disconnection of tournaments

The acyclic disconnection ω⃗(D) of a digraph D is defined as the maximum number of colors in a coloring of the vertices of D such that no cycle is properly colored (in a proper coloring, consecutive vertices of the directed cycle receive different colors). Similarly, the C⃗3-free disconnection ω⃗3(D) of D is the maximum number of colors in a coloring of the vertices of D such that no directed triangle is 3-colored. In this paper, we construct an infinite family Vn of tournaments T8n+1 with 8n+1 vertices (n∈N) such that ω⃗3(T8n+1)=n+2 and ω⃗(T8n+1)=2. This family allows us to solve the following problem posed by V. Neumann-Lara [V. Neumann-Lara, The acyclic disconnection of a digraph, Discrete Math. 197/198 (1999) 617-632]: Are there tournaments T for which ω⃗(T)=2 and ω⃗3(T) is arbitrarily large? The main result of the paper solves a generalization of the above problem: for positive integers r and s such that 2≤r≤s, there exists a tournament T such that ω⃗(T)=r and ω⃗3(T)=s.