Related papers: The diachromatic number of digraphs
The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph…
The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs…
A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be…
The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a…
We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the…
\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The…
The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$…
The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a $k$-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise non-adjacent vertices). The…
DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a…
The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising…
We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its…
We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number…
The dichromatic number $\vec{\chi}(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a…
The results of this note were stated in the first author PhD manuscript in 2006 but never published. The writing of a proof given there was slightly careless and the proof itself scattered across the document, the goal of this note is to…
The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic…
An acyclic coloring of a digraph as defined by Neumann-Lara is a vertex-coloring such that no monochromatic directed cycles occur. Counting the number of such colorings with $k$ colors can be done by counting so-called Neumann-Lara-coflows…
The dichromatic number of a digraph $G$ is the smallest integer $\chi_a(G)$ such that the vertex set of $G$ can be partitioned into $\chi_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic…
An acyclic coloring of a digraph that maximizes the number of colors such that each color class has a vertex pointing to all other classes and a vertex pointing to it from all other classes is known as the dib-chromatic number of a digraph.…
It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…
The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of…