Coloring Curves that Cross a Fixed Curve

Alexandre Rok,Bartosz Walczak

doi:10.1007/s00454-018-0031-z

Alexandre Rok, Bartosz Walczak

Open Access

https://doi.org/10.1007/s00454-018-0031-z

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Abstract

We prove that for every integer tgeqslant 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is chi -bounded. This is essentially the strongest chi -boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers kgeqslant 2 and tgeqslant 1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(nlog n) edges.

Highlights

OverviewA curve is a homeomorphic image of the real interval [0, 1] in the plane
The intersection graph of a family of curves has these curves as vertices and the intersecting
[20] showed that triangle-free intersection graphs of simple families of curves each crossing a fixed line in exactly one point have bounded chromatic number. Further progress in this direction was made by Suk [27], who proved that simple families of x-monotone curves crossing a fixed vertical line give rise to a χ -bounded class of intersection graphs, and by Lasonet al. [17], who reached the same conclusion without assuming that the curves are x-monotone

Summary

Introduction

A curve is a homeomorphic image of the real interval [0, 1] in the plane. The intersection graph of a family of curves has these curves as vertices and the intersecting. [20] showed that triangle-free intersection graphs of simple families of curves each crossing a fixed line in exactly one point have bounded chromatic number. Further progress in this direction was made by Suk [27], who proved that simple families of x-monotone curves crossing a fixed vertical line give rise to a χ -bounded class of intersection graphs, and by Lasonet al. In [26], we proved that the class of intersection graphs of curves each crossing a fixed line in exactly one point is χ -bounded.

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 6