A basic randomized coloring procedure has been used in probabilistic proofs to obtain remarkably strong results on graph coloring. These results include the asymptotic version of the List Coloring Conjecture due to Kahn, the extensions of Brooks’ Theorem to sparse graphs due to Kim and Johansson, and Luby’s fast parallel and distributed algorithms for graph coloring. The most challenging aspect...

A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way 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 ...

We study online interval coloring problems with bandwidth. We are interested in some variants motivated by bin packing problems. Specifically we consider open-end coloring, cardinality constrained coloring, coloring with vector constraints and finally a combination of both the cardinality and the vector constraints. We construct competitive algorithms for each of the variants. Additionally, we ...

Abstract Oriented coloring of an oriented graph G is arc-preserving homomorphism from into a tournament H . We say that the universal for family graphs $$\mathcal {C}$$ C if every $$G\in \mathcal G ∈ there exists are interested in finding orientations c...

The bandwidth coloring problem consists of assigning a color to each vertex of a graph, so that the absolute value of the difference between the colors of adjacent vertices is at least the value of the weight of the associated edge. This problem generalizes the classical vertex coloring problem and different heuristics have recently been proposed to obtain high quality solutions. In this paper ...

The eeciency of a parallel implementation of the conjugate gradient method precon-ditioned by an incomplete Cholesky factorization can vary dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we...

The eeciency of a parallel implementation of the conjugate gradient method precon-ditioned by an incomplete Cholesky factorization can vary dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we...

In this paper the new coloring of planar, VEF-coloring, will be introduced. A VEF coloring of a simple planar graph G is a proper coloring of all elements, including vertices, edges and faces of G. We will give two conjectures for the upper bound of VEF and VEF-list coloring of a simple planar graph. However, we will prove these conjectures for planar graphs with a maximum degree of at least 12...