The acyclic and -free disconnection of tournaments

The acyclic disconnection ω ⃗ ( 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 C ⃗ 3 -free disconnection ω ⃗ 3 ( 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 V n of tournaments T 8 n + 1 with 8 n + 1 vertices ( n ∈ N ) such that ω ⃗ 3 ( T 8 n + 1 ) = n + 2 and ω ⃗ ( T 8 n + 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 ω ⃗ ( T ) = 2 and ω ⃗ 3 ( 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 ω ⃗ ( T ) = r and ω ⃗ 3 ( T ) = s .