### Abstract

A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_{χ} with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + c_{χ} vertices of degree 3, where c_{χ} is a constant depending only on F_{χ}.

Original language | English |
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Pages (from-to) | 666-672 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 Mar 6 |

### Fingerprint

### Keywords

- 3-connected graph
- 3-tree
- Circuit graph
- Surface

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*309*(4), 666-672. https://doi.org/10.1016/j.disc.2008.01.002

**3-trees with few vertices of degree 3 in circuit graphs.** / Nakamoto, Atsuhiro; Oda, Yoshiaki; Ota, Katsuhiro.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 309, no. 4, pp. 666-672. https://doi.org/10.1016/j.disc.2008.01.002

}

TY - JOUR

T1 - 3-trees with few vertices of degree 3 in circuit graphs

AU - Nakamoto, Atsuhiro

AU - Oda, Yoshiaki

AU - Ota, Katsuhiro

PY - 2009/3/6

Y1 - 2009/3/6

N2 - A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

AB - A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

KW - 3-connected graph

KW - 3-tree

KW - Circuit graph

KW - Surface

UR - http://www.scopus.com/inward/record.url?scp=60149097255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60149097255&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2008.01.002

DO - 10.1016/j.disc.2008.01.002

M3 - Article

VL - 309

SP - 666

EP - 672

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -