License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.630
URN: urn:nbn:de:0030-drops-50915
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Felsner, Stefan ; Micek, Piotr ; Ueckerdt, Torsten

On-line Coloring between Two Lines

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We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(w^3) colors on graphs with maximum clique size w.

In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when w=2.

The left-of relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class C of posets obtained from convex sets between two lines with some other classes of posets: all 2-dimensional posets and all posets of height 2 are in C but there is a 3-dimensional poset of height 3 that does not belong to C.

We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.

BibTeX - Entry

  author =	{Stefan Felsner and Piotr Micek and Torsten Ueckerdt},
  title =	{{On-line Coloring between Two Lines}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{630--641},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-50915},
  doi =		{10.4230/LIPIcs.SOCG.2015.630},
  annote =	{Keywords: intersection graphs, cocomparability graphs, on-line coloring}

Keywords: intersection graphs, cocomparability graphs, on-line coloring
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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