### Abstract

A Grünbaum coloring of a triangulation G on a surface is a 3-edge coloring of G such that each face of G receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation on the projective plane P has a Grünbaum coloring, where a “Fisk triangulation” is one with exactly two odd degree vertices such that the two odd vertices are adjacent. To prove the theorem, we establish a generating theorem for Fisk triangulations on P. Moreover, we show that a triangulation G on P has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of G has even degree.

Original language | English |
---|---|

Pages (from-to) | 155-163 |

Number of pages | 9 |

Journal | Discrete Applied Mathematics |

Volume | 215 |

DOIs | |

State | Published - 2016 Dec 31 |

### Fingerprint

### Keywords

- Grünbaum coloring
- Projective plane
- Triangulation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*215*, 155-163. DOI: 10.1016/j.dam.2016.07.012

**Grünbaum colorings of triangulations on the projective plane.** / Kasai, Michiko; Matsumoto, Naoki; Nakamoto, Atsuhiro.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol 215, pp. 155-163. DOI: 10.1016/j.dam.2016.07.012

}

TY - JOUR

T1 - Grünbaum colorings of triangulations on the projective plane

AU - Kasai,Michiko

AU - Matsumoto,Naoki

AU - Nakamoto,Atsuhiro

PY - 2016/12/31

Y1 - 2016/12/31

N2 - A Grünbaum coloring of a triangulation G on a surface is a 3-edge coloring of G such that each face of G receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation on the projective plane P has a Grünbaum coloring, where a “Fisk triangulation” is one with exactly two odd degree vertices such that the two odd vertices are adjacent. To prove the theorem, we establish a generating theorem for Fisk triangulations on P. Moreover, we show that a triangulation G on P has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of G has even degree.

AB - A Grünbaum coloring of a triangulation G on a surface is a 3-edge coloring of G such that each face of G receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation on the projective plane P has a Grünbaum coloring, where a “Fisk triangulation” is one with exactly two odd degree vertices such that the two odd vertices are adjacent. To prove the theorem, we establish a generating theorem for Fisk triangulations on P. Moreover, we show that a triangulation G on P has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of G has even degree.

KW - Grünbaum coloring

KW - Projective plane

KW - Triangulation

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UR - http://www.scopus.com/inward/citedby.url?scp=84991030591&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2016.07.012

DO - 10.1016/j.dam.2016.07.012

M3 - Article

VL - 215

SP - 155

EP - 163

JO - Discrete Applied Mathematics

T2 - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -