### Abstract

We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.

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 | 28-39 |

Number of pages | 12 |

Volume | 8496 LNCS |

ISBN (Print) | 9783319078892 |

DOIs | |

State | Published - 2014 |

Event | 7th International Conference on Fun with Algorithms, FUN 2014 - Sicily, Italy Duration: Jul 1 2014 → Jul 3 2014 |

### 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 | 8496 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 7th International Conference on Fun with Algorithms, FUN 2014 |
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Country | Italy |

City | Sicily |

Period | 7/1/14 → 7/3/14 |

### 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. 8496 LNCS, pp. 28-39). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8496 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-07890-8_3

**Happy edges : Threshold-coloring of regular lattices.** / Alam, Md Jawaherul; Kobourov, Stephen G; Pupyrev, Sergey; Toeniskoetter, Jackson.

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. 8496 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8496 LNCS, Springer Verlag, pp. 28-39, 7th International Conference on Fun with Algorithms, FUN 2014, Sicily, Italy, 7/1/14. https://doi.org/10.1007/978-3-319-07890-8_3

}

TY - GEN

T1 - Happy edges

T2 - Threshold-coloring of regular lattices

AU - Alam, Md Jawaherul

AU - Kobourov, Stephen G

AU - Pupyrev, Sergey

AU - Toeniskoetter, Jackson

PY - 2014

Y1 - 2014

N2 - We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.

AB - We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.

UR - http://www.scopus.com/inward/record.url?scp=84903746702&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903746702&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-07890-8_3

DO - 10.1007/978-3-319-07890-8_3

M3 - Conference contribution

AN - SCOPUS:84903746702

SN - 9783319078892

VL - 8496 LNCS

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

SP - 28

EP - 39

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

PB - Springer Verlag

ER -