Publication

Triangle-free intersection graphs of line segments with large chromatic number

Bartosz Maria Walczak, Michal Lason
2014
Journal Articles

Abstract

In the 1970s Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer k we construct a triangle-free family of line segments in the plane with chromatic number greater than k. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number. (C) 2013 Elsevier Inc. All rights reserved.

Official source

https://infoscience.epfl.ch/entities/publication/a6297302-5754-4411-84d7-c920c1bbcacd

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Bartosz Maria Walczak, Michal Lason
2014
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/a6297302-5754-4411-84d7-c920c1bbcacd

About this result

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

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.

Intersection number (graph theory)

In the mathematical field of graph theory, the intersection number of a graph is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element. Equivalently, the intersection number is the smallest number of cliques needed to cover all of the edges of .

Chromatic polynomial

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics. George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem.

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