### 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 | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 26-37 |

Number of pages | 12 |

Volume | 8165 LNCS |

ISBN (Print) | 9783642450426 |

DOIs | |

State | Published - 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) | 03029743 |

ISSN (Electronic) | 16113349 |

### 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 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 8165 LNCS, 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

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 8165 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8165 LNCS, Springer Verlag, pp. 26-37, 39th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2013, Lubeck, Germany, 6/19/13. https://doi.org/10.1007/978-3-642-45043-3_4

}

TY - GEN

T1 - Threshold-coloring and unit-cube contact representation of graphs

AU - Alam, Md Jawaherul

AU - Chaplick, Steven

AU - Fijavž, Gašper

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Pupyrev, Sergey

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84893032513&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893032513&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-45043-3_4

DO - 10.1007/978-3-642-45043-3_4

M3 - Conference contribution

AN - SCOPUS:84893032513

SN - 9783642450426

VL - 8165 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 26

EP - 37

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -