Threshold-Coloring and Unit-Cube Contact Representation of Graphs

  • Md. Jawaherul Alam
  • Steven Chaplick
  • Gašper Fijavž
  • Michael Kaufmann
  • Stephen G. Kobourov
  • Sergey Pupyrev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)


We study threshold coloring of graphs where the vertex colors, represented by integers, describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. We show the NP-completeness for two variants of the threshold coloring problem and describe a polynomial-time algorithm for another.


Planar Graph Internal Vertex Graph Class Contact Representation Bottom Face 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • Steven Chaplick
    • 2
  • Gašper Fijavž
    • 3
  • Michael Kaufmann
    • 4
  • Stephen G. Kobourov
    • 1
  • Sergey Pupyrev
    • 1
    • 5
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  3. 3.Faculty of Computer and Information ScienceUniversity of LjubljanaSlovenia
  4. 4.Wilhelm-Schickhard-Institut für InformatikUniversität TübingenTübingenGermany
  5. 5.Institute of Mathematics and Computer ScienceUral Federal UniversityRussia

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