Two complexity results for the vertex coloring problem

Advice Complexity of the Online Vertex Coloring Problem

We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively. We show that, in the case of the first two classes, for coloring optimally, essentially log2 3 advice bits per vertex (bpv) are necessary and sufficient. In the...

Complexity and approximation results for the Min Weighted Coloring problem

A k-coloring of G = (V,E) is a partition S = (S1, . . . , Sk) of the node set V of G into stable sets Si (a stable set is a set of pairwise non adjacent nodes). In the usual case, the objective is to determine a node coloring minimizing k. A natural generalization of this problem is obtained by assigning a strictly positive integer weight w(v) for any node v ∈ V , and defining the weight of sta...

Polynomial Cases for the Vertex Coloring Problem

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for (P5, dart)-free graphs, (P5, banner)-free graphs, (P5, bull)-free graphs, and (fork, bull)-free graphs.

Complexity and Approximation Results for the Connected Vertex Cover Problem

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).

The Vertex Coloring Problem and its generalizations