### Abstract

In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that 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. Variants of the threshold-coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.

Original language | English (US) |
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Pages (from-to) | 2-14 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 216 |

DOIs | |

State | Published - Jan 10 2017 |

### Keywords

- Graph coloring
- Planar graphs
- Threshold-coloring
- Unit-cube contact representation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*216*, 2-14. https://doi.org/10.1016/j.dam.2015.09.003