Related papers: On Orthogonal Vector Edge Coloring

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…

The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph…

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number…

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…

Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of its ground set $X$. We give a combinatorial identity which relates the number of vectors in $S$ with given color patterns under $c$ with the cardinalities of the…

A total coloring of a simple undirected graph $G$ is an assignment of colors to its vertices and edges such that the colors given to the vertices form a proper vertex coloring, the colors given to the edges form a proper edge coloring, and…

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give $G$ a strong edge-coloring is called the strong…

The digraph chromatic number of a directed graph $D$, denoted $\chi_A(D)$, is the minimum positive integer $k$ such that there exists a partition of the vertices of $D$ into $k$ disjoint sets, each of which induces an acyclic subgraph. For…

Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$…

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…

Oriented chromatic number of an oriented graph $G$ is the minimum order of an oriented graph $H$ such that $G$ admits a homomorphism to $H$. The oriented chromatic number of an unoriented graph $G$ is the maximal chromatic number over all…

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in…

The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$…

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 {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same…

A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the…

Conflict-free coloring (in short, CF-coloring) of a graph $G = (V,E)$ is a coloring of $V$ such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on…

A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\chi_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all…

Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two…