In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by C...

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by C...

A majority coloring of a directed graph is vertex-coloring in which every vertex has the same color as at most half its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that digraph $4$-coloring conjectured admits $3$-coloring. They observed Local Lemma implies conjecture for digraphs large enough minimum out-degree if, crucially, maximum in-degree bounded by a(n exponential...

The λ-backbone coloring is one of the various problems of vertex colorings in graphs. Given an integer λ ≥ 2, a graph G = (V,E), and a spanning subgraph (backbone) H = (V, EH) of G, a λ-backbone coloring of (G,H) is a proper vertex coloring V → {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least λ. The λ-backbone coloring number BBCλ(G,H) of (G,H) is the s...

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most b and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes’ weights. In this paper we present complexity results and approximation algorithms for those problems on g...

of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs wit...