We revisit in this paper the probabilistic coloring problem ( ) and focus ourselves on bipartite and split graphs. We first give some general properties dealing with the optimal solution. We then show that the unique 2-coloring achieves approximation ratio 2 in bipartite graphs under any system of vertex-probabilities and propose a polynomial algorithm achieving tight appro...

We consider the following vertex coloring problem. We are given an undirected graph G = (V, E), where each vertex v is associated with a penalty rejection cost rv. We need to choose a subset of vertices, V ′, and to find a proper coloring of the induced subgraph of G over V ′. We are interested in both the number of colors used to color the vertices of V ′, and in the total rejection cost of al...

In the feedback vertex set problem, the aim is to minimize, in a connected graph G = (V ,E), the cardinality of the set V (G)⊆ V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in...

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring usin...

The problem of computing locally a coloring of an arbitrary planar subgraph of a unit disk graph is studied. Each vertex knows its coordinates in the plane, can directly communicate with all its neighbors within unit distance. Using this setting, first a simple algorithm is given whereby each vertex can compute its color in a 9-coloring of the planar graph using only information on the subgraph...

Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V , let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) 6= c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex co...

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 Hamming graphs, products of complete graphs, are considered.

An edge-coloring of a multigraph with colors 1, is called an interval coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that if is a connected cubic multigraph (a connected cubic graph) that admits an interval coloring, then G 2, , t ... t −

Efficiently coloring an arbitrary graph is a fundamental and notoriously difficult algorithmic problem. This talk focuses on the restricted problem of determining the complexity of coloring graphs which do not contain a certain induced subgraph. Combining results of Kamiński and Lozin, and Hoyler, it follows that this problem remains NP-complete unless the excluded induced subgraph is a disjoin...