相关论文: Bounding the chromatic number of dense digraphs by…
The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the…
The dichromatic number $\chi(\vec{G})$ of a digraph $\vec{G}$ is the minimum number of colors needed to color the vertices $V(\vec{G})$ in such a way that no monochromatic directed cycle is obtained. In this note, for any $k\in \mathbb{N}$,…
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the…
Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$.…
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…
We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$.…
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 local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when…
An arc-coloured digraph $D$ is said to be \emph{rainbow connected} if for every two vertices $u$ and $v$ there is an $uv$-path all whose arcs have different colours. The minimun number of colours required to make the digraph rainbow…
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 $G$…
A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if…
Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs…
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…
We study an extension to directed graphs of the parameter called the $b$-chromatic number of a graph in terms of acyclic vertex colorings: the dib-chromatic number. We give general bounds for this parameter. We also show some results about…
A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss…
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vec{\chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph…
The dichromatic number $\vec\chi(D)$ of a digraph is the minimum number $k$ such that $V(D)$ can be partitioned into $k$ subsets, each inducing an acyclic digraph. The acyclic number $\vec\alpha(D)$ is the cardinality of a largest induced…
This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments A,B, both with large chromatic…
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…
\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…