Related papers: List coloring digraphs
The dichromatic number of $D$, denoted by $\overrightarrow{\chi}(D)$, is the smallest integer $k$ such that $D$ admits an acyclic $k$-coloring. We use $mader_{\overrightarrow{\chi}}(F)$ to denote the smallest integer $k$ such that if…
We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list…
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…
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…
DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo\v{r}\'{a}k and Postle. Several known bounds for the list chromatic number of a graph $G$, $\chi_\ell(G)$, also hold for the…
A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a…
A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is…
The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…
An $r$-dynamic $k$-coloring of a graph $G$ is a proper vertex $k$-coloring such that the neighbors of any vertex $v$ receive at least $\min\{r,{\rm deg}(v)\}$ different colors. The $r$-dynamic chromatic number of $G$, $\chi_r(G)$, is…
We prove that every triangle-free graph with maximum degree $\Delta$ has list chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that…
We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have…
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than…
A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of…
Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of…
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…
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…
The studies on $b$-chromatic number attracted much interest since its introduction. In this paper, we discuss the $b$-chromatic number of certain classes of graphs and digraphs. The notion of a new general family of graphs called the…
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…
DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that…
Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order…