A mixed hypergraph H = (X, C,D) consists of the vertex set X and two families of subsets: the family C of C-edges and the family D of D-edges. In a coloring, every C-edge has at least two vertices of common color, while every D-edge has at least two vertices of different colors. The largest (smallest) number of colors for which a coloring of a mixed hypergraph H using all the colors exists is c...

Let S = (a1, a2, . . .) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V (G) to {1, 2, . . . , k} such that vertices with color i have pairwise distance greater than ai, and the S-packing chromatic number χS(G) of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper...

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G is said k-choosable. A graph is said to be acyclically k-...

A harmonious coloring of a k-uniform hypergraphH is a rainbow vertex coloring such that each k-set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k-set of colors appears on at least one edge. The harmonious (resp. achromatic) number of H , denoted by h(H) (resp. ψ(H)) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic...

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of products of paths and cycles are considered. We determine the acyclic chromatic numbers of three such products: grid graphs...

A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, χD(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring....

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1 − 1 k of edges. We prove that given a graph promised to be k-colorable, it is N...

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every...

A strong edge coloring of a graph G is an edge coloring such that every two adjacent edges or two edges adjacent to a same edge receive two distinct colors; in other words, every path of length three has three distinct colors in G. The strong chromatic index of G, denoted by   S G  , is the smallest integer k such that G admits a strong edge coloring with k colors. This survey is an brief i...