New Bounds for the Dichromatic Number of a Digraph

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 dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at most its circumference, the length of a longest cycle. Given a digraph $D$, we will construct three different graphs whose chromatic numbers bound $\chi_A(D)$. Moreover, we prove: i) for integers $k\geq 2$, $s\geq 1$ and $r_1, \ldots, r_s$ with $k\geq r_i\geq 0$ and $r_i\neq 1$ for each $i\in[s]$, that if all cycles in $D$ have length $r$ modulo $k$ for some $r\in\{r_1,\ldots,r_s\}$, then $\chi_A(D)\leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$ and $p$, with $k\geq g-1\geq p\geq 1$ such that $D$ contains no cycle of length $r$ modulo $\lceil \frac{k}{p} \rceil p$ for each $r\in \{-p+2,\ldots,0,\ldots,p\}$, then $\chi_A (D)\leq \lceil \frac{k}{p} \rceil$; iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference $c$, then $\chi_A(D)\leq \lceil \frac{c-1}{g-1} \rceil +1$, which improves, substantially, the bound proposed by Bondy. Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now.Comment: 14 page

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

On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

The A_α spectral moments of digraphs with a given dichromatic number

The Aα-matrix of a digraph G is defined as Aα(G)=αD+(G)+(1−α)A(G), where α∈[0,1), D+(G) is the diagonal outdegree matrix and A(G) is the adjacency matrix. The k-th Aα spectral moment of G is defined as ∑i=1 nλαi k, where λαi are the eigenvalues of the Aα-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second Aα spectral moment (also known as the Aα energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third Aα spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs.</p