### 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 |
---|---|

Pages (from-to) | 519-524 |

Number of pages | 6 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 11 |

DOIs | |

Publication status | Published - 2002 Jul |

### Fingerprint

### Keywords

- 3-coloring
- plane triangulation
- triangulation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

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

**On 3-coloring of plane triangulations.** / Nakamoto, Atsuhiro; Ota, Katsuhiro; Watanabe, Mamoru.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - On 3-coloring of plane triangulations

AU - Nakamoto, Atsuhiro

AU - Ota, Katsuhiro

AU - Watanabe, Mamoru

PY - 2002/7

Y1 - 2002/7

N2 - 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 hλ (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 {hλ (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.

AB - 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 hλ (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 {hλ (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.

KW - 3-coloring

KW - plane triangulation

KW - triangulation

UR - http://www.scopus.com/inward/record.url?scp=34247178092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247178092&partnerID=8YFLogxK

U2 - 10.1016/S1571-0653(04)00097-6

DO - 10.1016/S1571-0653(04)00097-6

M3 - Article

AN - SCOPUS:34247178092

VL - 11

SP - 519

EP - 524

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -