The Dichromatic Number of Infinite Families of Circulant Tournaments

• Autorzy: Nahid Javier , Bernardo Llano
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

tournament; dichromatic number; vertex-critical r-dichromatic tournament

Kategorie tematyczne

• 05C20: Directed graphs (digraphs), tournaments
• 05C38: Paths and cycles

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2017
• Tom: 37
• Numer: 1
• Strony: 221-238

Daty

wydano

2017-02-01

poprawiono

2016-04-07

zaakceptowano

2016-04-07

otrzymano

2016-06-30

online

2017-01-13

Twórcy

autor

Nahid Javier

• Departamento de Matemáticas, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, Colonia Vicentina, 09340, México, D.F.,

autor

Bernardo Llano

• Departamento de Matemáticas, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, Colonia Vicentina, 09340, México, D.F.,

Identyfikatory

DOI

10.7151/dmgt.1930