### Abstract

Let G be a plane triangulation. For a 3-vertex-coloring λ, a face of G whose vertices receive all three colors is called a vivid face with respect to λ. Let h_{λ} (G) be the number of vivid faces in G with respect to λ. Let C(G) be the set of 3-vertex-colorings of G and let g(n) be the set of plane triangulations with n faces. Let h(G) = max {h_{λ} (G) {divides} λ ∈ C(G)} and h(n) = min {h(G) {divides} G ∈ g(n)}. In this paper we show that h(n) ≥ 1 2 n for any even n, and that h(n) ≤ 1 5 (3n - 2) for infinitely many n.

Original language | English |
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Pages (from-to) | 519-524 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 11 |

DOIs | |

Publication status | Published - 2002 Jul 1 |

### Keywords

- 3-coloring
- plane triangulation
- triangulation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

Nakamoto, A., Ota, K., & Watanabe, M. (2002). On 3-coloring of plane triangulations.

*Electronic Notes in Discrete Mathematics*,*11*, 519-524. https://doi.org/10.1016/S1571-0653(04)00097-6