### 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 | SOFSEM 2015 |

Subtitle of host publication | Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science, |

Editors | Tiziana Margaria-Steffen, Giuseppe F. Italiano, Jean-Jacques Quisquater, Roger Wattenhofer, Jaroslav Pokorný |

Publisher | Springer-Verlag |

Pages | 115-127 |

Number of pages | 13 |

ISBN (Electronic) | 9783662460771 |

State | Published - Jan 1 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) |
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Volume | 8939 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

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

City | Pec pod Sněžkou |

Period | 1/24/15 → 1/29/15 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*SOFSEM 2015: Theory and Practice of Computer Science - 41st International Conference on Current Trends in Theory and Practice of Computer Science,*(pp. 115-127). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8939). Springer-Verlag.