### Abstract

Let G be a 3-connected graph with n vertices on a non-spherical closed surface F_{k}^{2} of Euler genus k with sufficiently large representativity. In this paper, we first study a new cutting method which produces a spanning planar subgraph of G with a certain good property. This is used to show that such a graph G has a spanning 4-tree with at most max{2k - 5, 0} vertices of degree 4. Using this result, we prove that for any integer t, if n is sufficiently large, then G has a connected subgraph with t vertices whose degree sum is at most 8t - 1. We also give a nearly sharp bound for the projective plane, torus and Klein bottle. Furthermore, using our cutting method, we prove that a 3-connected graph G on F_{k}^{2} with high representativity has a 3-walk in which at most max{2k - 4, 0} vertices are visited three times, and an 8-covering with at most max{4k - 8, 0} vertices of degree 7 or 8. Moreover, a 4-connected G has a 4-covering with at most max[4k - 6, 0} vertices of degree 4.

Original language | English |
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Pages (from-to) | 207-229 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 89 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 Nov |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*89*(2), 207-229. https://doi.org/10.1016/S0095-8956(03)00072-8