In this paper, we give bounds on the dichromatic number $\vecχ(Σ)$ of a surface $Σ$, which is the maximum dichromatic number of an oriented graph embeddable on $Σ$. We determine the asymptotic behaviour of $\vecχ(Σ)$ by showing that there exist constants $a_1$ and $a_2$ such that, $a_1\frac{\sqrt{-c}}{\log(-c)} \leq \vecχ(Σ) \leq a_2 \frac{\sqrt{-c}}{\log(-c)} $ for every surface $Σ$ with Euler characteristic $c\leq -2$. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane $\mathbb{N}_1$, the Klein bottle $\mathbb{N}_2$, the torus $\mathbb{S}_1$, and Dyck's surface $\mathbb{N}_3$ are all equal to $3$, and that the dichromatic numbers of the $5$-torus $\mathbb{S}_5$ and the $10$-cross surface $\mathbb{N}_{10}$ are equal to $4$. We also consider the complexity of deciding whether a given digraph or oriented graph embeddable on a fixed surface is $k$-dicolourable. In particular, we show that for any fixed surface, deciding whether a digraph embeddable on this surface is $2$-dicolourable is NP-complete, and that deciding whether a planar oriented graph is $2$-dicolourable is NP-complete unless all planar oriented graphs are $2$-dicolourable (which was conjectured by Neumann-Lara).
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree T and every integer k, any oriented graph that does not contain an induced copy of T nor a clique of size k has dichromatic number at most some function of k and T.
The dichromatic number $\vecχ(D)$ 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 authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph $D$ is $3$-dicritical if the removal of any arc of $D$ reduces the dichromatic number to 2. We construct infinitely many $3$-dicritical oriented graphs. Neumann-Lara found the four $7$-vertex $3$-dichromatic tournaments. We determine the $8$-vertex $3$-dichromatic tournaments, which …
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 directed cycle under this orientation. In 1979, Erdős and Neumann-Lara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We make the first significant progress on this conjecture by proving a fractional version of the conjecture. While our result uses stronger assumption about the fractional chromatic number, it also …
We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.
We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vecχ^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are "easier" or "harder" to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced by Wang et al. and resembles the concept of the star chromatic number of graphs introduced by Vince in the framework of digraph colouring. After presenting basic properties of the new quantity, includin…
The dichromatic number $\vecχ(D)$ 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 invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number…
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vecχ}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is tha…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we…
Chris Lambie-Hanson proved recently that for every function $ f:\mathbb{N}\rightarrow \mathbb{N} $ there is an $ \aleph_1 $-chromatic graph $ G $ of size $ 2^{\aleph_1} $ such that every $ (n+3) $-chromatic subgraph of $ G $ has at least $ f(n) $ vertices. Previously, this fact was just known to be consistently true due to P. Komjáth and S. Shelah. We investigate the analogue of this question for directed graphs. In the first part of the paper we give a simple method to construct for an arbitrary $ f:\mathbb{N}\rightarrow \mathbb{N} $ an uncountably dichromatic digraph $ D $ of size $ 2^{\alep…
We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomassé.
Motivated by an old conjecture of P. Erdős and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that consistently there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected ca…
We propose a purely combinatorial quadratic time algorithm that for any $n$-vertex $P_{k}$-free tournament $T$, where $P_{k}$ is a directed path of length $k$, finds in $T$ a transitive subset of order $n^{\frac{c}{k\log(k)^{2}}}$. As a byproduct of our method, we obtain subcubic $O(n^{1-\frac{c}{k\log(k)^{2}}})$-approximation algorithm for the optimal acyclic coloring problem on $P_{k}$-free tournaments. Our results are tight up to the $\log(k)$-factor in the following sense: there exist infinite families of $P_{k}$-free tournaments with largest transitive subsets of order at most $n^{\frac{c…
Let $G$ be a graph of order $n$. It is well-known that $α(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $α(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ α(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $α(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $χ(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(Δ^+(D),Δ^-(D))$. We give a short and simple proof for…
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…
The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…
For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced directed cycles in $D$ have length equal to $t$. In a previous paper, we asked for which $t$ it is true that $t$-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for $t=3$, that is, they gave a construction of $3$-chordal digraphs with clique number at most $3$ and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each $t \ge 3$ a construction of digraphs with clique number at most $3$…
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyárfás-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets $\mathcal F$ of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induce subdigraph. Among notable…
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 dichroma…
In 1982 Víctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow theory for the dichromatic number transferring Tutte's theory of nowhere-zero flows (NZ-flows) from classic graph colorings has been developed by Hochstättler. The purpose of this paper is to pursue this analogy by introducing a new definition of algebraic Neumann-Lara-flows (NL-flows) and a closed formula for their polynomial. Furthermore we generalize the Equiva…
The dichromatic number of $D$, denoted by $\overrightarrowχ(D)$, is the smallest integer $k$ such that $D$ admits an acyclic $k$-coloring. We use $mader_{\overrightarrowχ}(F)$ to denote the smallest integer $k$ such that if $\overrightarrowχ(D)\ge k$, then $D$ contains a subdivision of $F$. A digraph $F$ is called Mader-perfect if for every subdigraph $F'$ of $F$, ${\rm mader }_{\overrightarrowχ}(F')=|V(F')|$. We extend octi digraphs to a larger class of digraphs and prove that it is Mader-perfect, which generalizes a result of Gishboliner, Steiner and Szabó [Dichromatic number and forced subd…
The dichromatic number of a digraph $D$ is the minimum number of colors of a vertex coloring of $D$ such that $D$ has no monochromatic cycles. The Hajós join were recently extended to digraphs (using the dichromatic number) by J. Bang-Jensen et. al. and Hajós (directed) operations is a tool to obtain r-(di)chromatic (di)graphs. J. Bang-Jensen et. al. posed in 2020 the problem of how to obtain the symmetric cycle of length 5 from symmetric cycles of length 3. We recently solved this problem by applying a genetic algorithm. In this article, a procedure is presented to construct any odd symmetric…
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 diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation property and the Nordhaus-Gaddum relations for the digrundy number, and improve the Nordhaus-Gaddum relations for the dichromatic and diachromatic numbers bounded previously by the authors in [Electron. J. Combin. 25 (2018) no. 3, Paper {\#} 3.51, 17 pp.]
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.
We prove that for every $n$, there is a graph $G$ with $χ(G) \geq n$ and $ω(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $ω(H) \leq 2$ satisfies $χ(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, 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 \emph{dichromatic number} $χ_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $χ_d(D)$ analogous to that of chromatic number of a graph, we prove $χ_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained …
A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positi…
The dichromatic number $\dic(D)$ of a digraph $D$ is the least integer $k$ such that $D$ can be partitioned into $k$ directed acyclic digraphs. A digraph is $k$-dicritical if $\dic(D) = k$ and each proper subgraph $D'$ of $D$ satisfies $\dic(D') \leq k-1$. An oriented graph is a digraph with no directed cycle of length $2$. For integers $k$ and $n$, we denote by $o_k(n)$ the minimum number of edges of a $k$-critical oriented graph on $n$ vertices (with the convention $o_k(n)=+\infty$ if there is no $k$-dicritical oriented graph of order $n$). The main result of this paper is a proof that $o_3(…
