Publication

Conflict-free colorings of graphs and hypergraphs Probability and Computing

2009
Journal Articles

Abstract

A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of 'unique' colour that does not get repeated in E. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by chi(CF)(H). This parameter wits first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that chi(CF)(H) = 3. Using Lovasz's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t - 1 and every edge intersects at most tit others. We give efficient polynomial-time algorithms to obtain such colourings.

Official source

https://infoscience.epfl.ch/entities/publication/08bd6a9e-b250-4dc5-9499-3f69fcb1962a

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2009
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/08bd6a9e-b250-4dc5-9499-3f69fcb1962a

About this result

Related concepts (23)

Graph coloring

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Greedy coloring

In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible. Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering.

Fractional coloring

Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common.

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Conflict-free colorings of graphs and hypergraphs Probability and Computing

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