### Abstract

Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (⌊2n/3⌋, 2n)-differential coloring problem, even in the case where the input graph is planar.

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 | 115-127 |

Number of pages | 13 |

Volume | 8939 |

ISBN (Print) | 9783662460771 |

State | Published - 2015 |

Event | 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015 - Pec pod Sněžkou, Czech Republic Duration: Jan 24 2015 → Jan 29 2015 |

### Publication series

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

Volume | 8939 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015 |
---|---|

Country | Czech Republic |

City | Pec pod Sněžkou |

Period | 1/24/15 → 1/29/15 |

### 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. 8939, pp. 115-127). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8939). Springer Verlag.

**The maximum k-differential coloring problem.** / Bekos, Michael A.; Kaufmann, Michael; Kobourov, Stephen G; Veeramoni, Sankar.

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. 8939, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8939, Springer Verlag, pp. 115-127, 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015, Pec pod Sněžkou, Czech Republic, 1/24/15.

}

TY - GEN

T1 - The maximum k-differential coloring problem

AU - Bekos, Michael A.

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Veeramoni, Sankar

PY - 2015

Y1 - 2015

N2 - Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (⌊2n/3⌋, 2n)-differential coloring problem, even in the case where the input graph is planar.

AB - Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (⌊2n/3⌋, 2n)-differential coloring problem, even in the case where the input graph is planar.

UR - http://www.scopus.com/inward/record.url?scp=84922051449&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922051449&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84922051449

SN - 9783662460771

VL - 8939

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

SP - 115

EP - 127

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

PB - Springer Verlag

ER -