Complete Acyclic Colorings

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 vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

The complexity of deciding whether a graph admits an orientation with fixed weak diameter

International audienceAn oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems

Complete oriented colourings and the oriented achromatic number

International audienceIn this paper, we initiate the study of %complete homomorphisms, or complete colourings of oriented graphs and the new associated notion of the oriented achromatic number of oriented and undirected graphs. In particular, we prove that for every integers $a$ and $b$ with $2\le a\le b$, there exists an oriented graph \vv{G}\!_{a,b} with oriented chromatic number $a$ and oriented achromatic number $b$. We also prove that adding a vertex, deleting a vertex or deleting an arc in an oriented graph may increase its oriented achromatic number by an arbitrarily large value, while adding an arc may increase its oriented achromatic number by at most 2. Finally, we consider the behaviour of the oriented chromatic and achromatic numbers with respect to elementary homomorphisms and show in particular that, in contrast to the undirected case, there is no interpolation homomorphism theorem for oriented graphs. Our notion of complete colouring of oriented graphs corresponds to the notion of complete homomorphisms of oriented graphs and, therefore, differs from the notion of complete colourings of directed graphs recently introduced by Edwards in [Harmonious chromatic number of directed graphs. {\em Discrete Appl. Math.} {\bf 161} (2013), 369--376.]