Related papers: On DP-Coloring of Digraphs

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…

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a…

Defective coloring (also known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of…

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…

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…

We define $Z$-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for…

Motivated by an old conjecture of P. Erd\H{o}s 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…

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are…

The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for…

A \emph{request} on a graph assigns a preferred color to a subset of the vertices. A graph $G$ is \emph{$\epsilon$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that…

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound $\chi(G) \leq (1+o(1))\Delta(G)/\ln \Delta(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy…

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly $f$-degenerate transversal, is a further generalization of DP-coloring and $L$-forested-coloring. In…

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph $G=(V,E)$, an oriented $r$-coloring for $G$ is a partition of the vertex set $V$ into $r$ independent sets, such…

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…

Let $D=(V,A)$ be a digraph. We define $\Delta_{\max}(D)$ as the maximum of $\{ \max(d^+(v),d^-(v)) \mid v \in V \}$ and $\Delta_{\min}(D)$ as the maximum of $\{ \min(d^+(v),d^-(v)) \mid v \in V \}$. It is known that the dichromatic number…

Classical problems in hypergraph coloring theory are to estimate the minimum number of edges, $m_2(r)$ (respectively, $m^\ast_2(r)$), in a non-$2$-colorable $r$-uniform (respectively, $r$-uniform and simple) hypergraph. The best currently…

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…

It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…

The dichromatic number $\vec{\chi}(D)$ 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…