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 journalArticlepeer-review


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)
Pages (from-to)2-14
Number of pages13
JournalDiscrete Applied Mathematics
StatePublished - Jan 10 2017


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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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