The Dichromatic Number of Infinite Families of Circulant Tournaments

Nahid Javier ; Bernardo Llano

Discussiones Mathematicae Graph Theory, Tome 37 (2017), p. 221-238 / Harvested from The Polish Digital Mathematics Library

Résumé

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 \lbrace 2,3,4\rbrace $ 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.

Détail

Publié le : 2017-01-01

Zbl 1354.05059

EUDML-ID : urn:eudml:doc:287990