### 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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Journal | Discrete Applied Mathematics |

DOIs | |

State | Accepted/In press - Apr 16 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2015.09.003

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

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*. https://doi.org/10.1016/j.dam.2015.09.003

}

TY - JOUR

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

AU - Alam, Md Jawaherul

AU - Chaplick, Steven

AU - Fijavž, Gašper

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Pupyrev, Sergey

AU - Toeniskoetter, Jackson

PY - 2014/4/16

Y1 - 2014/4/16

N2 - 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.

AB - 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.

KW - Graph coloring

KW - Planar graphs

KW - Threshold-coloring

KW - Unit-cube contact representation

UR - http://www.scopus.com/inward/record.url?scp=84950145059&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950145059&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2015.09.003

DO - 10.1016/j.dam.2015.09.003

M3 - Article

AN - SCOPUS:84950145059

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -