Related papers: Neighbor-Locating Colorings in Graphs
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…
A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the two sets of colors found in their respective neighborhoods are different. The…
Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,...,V_k)$ 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…
An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…
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…
For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that each vertex has an equal number of neighbors of each color is called neighborhood-balanced…
A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further…
For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that every vertex has an equal number of vertices of each color in its closed neighborhood is called…
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…
For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors…
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 $k$.…
Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,\ldots,V_k)$ 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…
A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a…
A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $\varphi:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $\varphi(u)=\varphi(v)$ are in distance at least $\varphi(u)+1$. This concept is motivated…
For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored…
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…
Generalizing the notion of odd-sum colorings, a $\mathbb{Z}$-labeling of a graph $G$ is called a closed coloring with remainder $k\mod n$ if the closed neighborhood label sum of each vertex is congruent to $k\mod n$. If such colorings…
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…
A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using…