Related papers: Conditional and Unique Coloring of Graphs

A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that…

We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such…

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the…

Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies that vertices $u$ and $v$ are at distance greater than $i$, is called a packing coloring of $G$. The minimum number of colors in a packing…

A proper vertex coloring of a graph $G$ is $r$-dynamic if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. In this paper we investigate $r$-dynamic versions of coloring, list coloring, and paintability. We prove…

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. Let $G$ and $H$ be $r$-graphs. An $H$-coloring of $G$ is a mapping $f\colon E(G) \to E(H)$ such that each $r$ adjacent…

A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{neighbor-locating} if for every pair of vertices $u,v$ belonging to the same…

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv$ of $G$. The smallest integer $k$ for which there is a proper $k$-coloring…

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.…

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$.…

In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$. A…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing)…

A $vi$-simultaneous proper $k$-coloring of a graph $G$ is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set $V(G)\cup I(G)$ receive distinct colors, where $I(G)$ is the set of…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

A $b$-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the maximum…

Let $c$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be…

Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$, the ($d,kn$)-differential coloring problem asks for a coloring of the vertices of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as…

A $k$-coloring of a graph $G$ is a partition of the set of vertices of $G$ into $k$ independent sets, which are called colors. A $k$-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from…

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. It is proved that in…