Threshold-coloring and unit-cube contact representation of planar graphs

Md Jawaherul Alam, Steven Chaplick, Gašper Fijavž, Michael Kaufmann, Stephen G Kobourov, Sergey Pupyrev, Jackson Toeniskoetter

Research output: Contribution to journalArticle

Abstract

In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that 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. Variants of the threshold-coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.

Original languageEnglish (US)
JournalDiscrete Applied Mathematics
DOIs
StateAccepted/In press - Apr 16 2014

Fingerprint

Unit cube
Coloring
Planar graph
Colouring
Contact
Color
Graph in graph theory
Grid Graph
Imply
Spanning Subgraph
NP-completeness
Graph Coloring
Polynomials
Polynomial-time Algorithm
Cycle
Integer
Vertex of a graph

Keywords

  • Graph coloring
  • Planar graphs
  • Threshold-coloring
  • Unit-cube contact representation

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Alam, M. J., Chaplick, S., Fijavž, G., Kaufmann, M., Kobourov, S. G., Pupyrev, S., & Toeniskoetter, J. (Accepted/In press). Threshold-coloring and unit-cube contact representation of planar graphs. Discrete Applied Mathematics. https://doi.org/10.1016/j.dam.2015.09.003

Threshold-coloring and unit-cube contact representation of planar graphs. / Alam, Md Jawaherul; Chaplick, Steven; Fijavž, Gašper; Kaufmann, Michael; Kobourov, Stephen G; Pupyrev, Sergey; Toeniskoetter, Jackson.

In: Discrete Applied Mathematics, 16.04.2014.

Research output: Contribution to journalArticle

Alam, Md Jawaherul ; Chaplick, Steven ; Fijavž, Gašper ; Kaufmann, Michael ; Kobourov, Stephen G ; Pupyrev, Sergey ; Toeniskoetter, Jackson. / Threshold-coloring and unit-cube contact representation of planar graphs. In: Discrete Applied Mathematics. 2014.
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