Polychromatic Colorings of Plane Graphs.

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Bibliographic Details
Title:	Polychromatic Colorings of Plane Graphs.
Authors:	Alon, Noga¹ nogaa@tau.ac.il, Berke, Robert² berker@ethz.ch, Buchin, Kevin³ buchin@cs.uu.nl, Buchin, Maike³ maike@cs.uu.nl, Csorba, Péter⁴ pcsorba@win.tue.nl, Shannigrahi, Saswata⁵ saswata@tcs.tifr.res.in, Speckmann, Bettina⁴ speckman@win.tue.nl, Zumstein, Philipp⁶ zuphilip@inf.ethz.ch
Source:	Discrete & Computational Geometry. Oct2009, Vol. 42 Issue 3, p421-442. 22p. 9 Diagrams.
Subjects:	Polychromy, Central processing units, Dynamic random access memory, Computer input-output equipment, Computer storage device industry, Computer systems, Computer networks
Abstract:	We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ⌊(3 g−5)/4 ⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3 g+1)/4 ⌋ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in ℘ for k=2 and is $\mathcal{NP}$ -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in ℘, and for the others it is $\mathcal{NP}$ -complete. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

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Abstract:	We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ⌊(3 g−5)/4 ⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3 g+1)/4 ⌋ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in ℘ for k=2 and is $\mathcal{NP}$ -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in ℘, and for the others it is $\mathcal{NP}$ -complete. [ABSTRACT FROM AUTHOR]
ISSN:	01795376
DOI:	10.1007/s00454-009-9171-5