The Dichromatic Number of Infinite Families of Circulant Tournaments

The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by T=C→2n+1(1,2,…,n)$T = \overrightarrow C _{2n + 1} (1,2, \ldots ,n)$, where V (T) = ℤ2n+1 and for every jump j ∈ {1, 2, . . . , n} there exist the arcs (a, a + j) for every a ∈ ℤ2n+1. Consider the circulant tournament C→2n+1〈k〉$\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ obtained from the cyclic tournament by reversing one of its jumps, that is, C→2n+1 〈k〉$\overrightarrow C _{2n + 1} \left\langle k \right\rangle $ has the same arc set as C→2n+1(1,2,…,n)$\overrightarrow C _{2n + 1} (1,2, \ldots ,n)$ except for j = k in which case, the arcs are (a, a − k) for every a ∈ ℤ2n+1. In this paper, we prove that dc(C→2n+1 〈k〉)∈{2,3,4}$dc ( {\overrightarrow C _{2n + 1} \left\langle k \right\rangle } ) \in \{ 2,3,4\}$ for every k ∈ {1, 2, . . . , n}. Moreover, we classify which circulant tournaments C→2n+1 〈k〉$\overrightarrow C _{2n + 1} \left\langle k \right\rangle$ are vertex-critical r-dichromatic for every k ∈ {1, 2, . . . , n} and r ∈ {2, 3, 4}. Some previous results by Neumann-Lara are generalized