### 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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Pages (from-to) | 35-53 |

Number of pages | 19 |

Journal | Journal of Discrete Algorithms |

Volume | 45 |

DOIs | |

Publication status | Published - Jul 1 2017 |

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### Keywords

- Differential chromatic number
- Differential coloring
- Maximum k-differential coloring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Discrete Algorithms*,

*45*, 35-53. https://doi.org/10.1016/j.jda.2017.08.001