We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes x, y, there is an arc from x to y and an arc from y to x. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.
In this paper, we construct infinite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an infinite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show that any tournament with a tight mold is also tight.
A rainbow t-coloring of a t-connected graph G is an edge coloring such that for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u,v)-paths. In this work, we apply a Rank Genetic Algorithm to search for rainbow t-colorings of the family of Moore cages with girth six (t;6)-cages. We found that an upper bound in the number of colors needed to produce a rainbow 4-coloring of a (4;6)-cage is 7, improving the one currently known, which is 13. The computation of the minimum number of colors of a rainbow coloring is known to be NP-Hard and the Rank Genetic Algorithm showed good behavior finding rainbow t-colorings with a small number of colors.
We disprove the following conjecture due to Víctor Neumann-Lara: for every pair $(r,s)$ of integers such that $r\geq s\geq 2$, there is an infinite set of circulant tournaments $T$ such that the dichromatic number and the cyclic triangle free disconnection of $T$ are equal to $r$ and $s$, respectively. Let $\mathcal{F}_{r,s}$ denote the set of circulant tournaments $T$ with $dc(T)=r$ and $\overrightarrow{\omega }_{3}\left( T\right) =s$. We show that for every integer $s\geq 4$ there exists a lower bound $b(s)$ for the dichromatic number $r$ such that $\mathcal{F}_{r,s}=\emptyset $ for every $r<b(s)$. We construct an infinite set of circulant tournaments $T$ such that $dc(T)=b(s)$ and $\overrightarrow{\omega }_{3}(T)=s$ and give an upper bound $B(s)$ for the dichromatic number $r$ such that for every $r\geq B(s)$ there exists an infinite set $\mathcal{F}_{r,s}$ of circulant tournaments. Some infinite sets $\mathcal{F}_{r,s}$ of circulant tournaments are given for $b(s)<r<B(s)$.
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