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 every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list a...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list tot...

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring...

We investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) has applications, among other areas, in efficient solving of sparse linear systems of equations and scheduling parallel assembly of products. Our main technical results improve upper and lower bounds for the ordered chromatic number of...

the robust coloring problem (rcp) is a generalization of the well-known graph coloring problem where we seek for a solution that remains valid when extra edges are added. the rcp is used in scheduling of events with possible last-minute changes and study frequency assignments of the electromagnetic spectrum. this problem has been proved as np-hard and in instances larger than 30 vertices, meta-...

An incidence of G consists of a vertex and one of its incident edge in G. The incidence coloring problem is a variation of vertex coloring problem. The problem is to find the minimum number (called incidence coloring number) of colors assigned to every incidence of G so that the adjacent incidences are not assigned the same color. In this paper, we propose a linear time algorithm for incidence-...