### Abstract

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.

Original language | English (US) |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 39th International Workshop, WG 2013, Revised Papers |

Publisher | Springer-Verlag |

Pages | 26-37 |

Number of pages | 12 |

ISBN (Print) | 9783642450426 |

DOIs | |

State | Published - Jan 1 2013 |

Event | 39th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2013 - Lubeck, Germany Duration: Jun 19 2013 → Jun 21 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8165 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 39th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2013 |
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Country | Germany |

City | Lubeck |

Period | 6/19/13 → 6/21/13 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph-Theoretic Concepts in Computer Science - 39th International Workshop, WG 2013, Revised Papers*(pp. 26-37). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8165 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-642-45043-3_4