### Abstract

Let c(H) denote the number of components of a graph H. Win proved in 1989 that if a connected graph G satisfies c(G\S)≤(k- 2)|S| + 2, for every subset S of V(G), then G has a spanning tree with maximum degree at most k. For a spanning tree T of a connected graph, the k-excess of a vertex v is defined to be max{0, deg_{T}(v) -k}. The total k-excess te(T, k) is the summation of the k-excesses of all vertices, namely, te(T, k) =Σ _{vεV(T)}max{0. deg_{T}(v) - k}. This paper gives a sufficient condition for a graph to have a spanning tree with bounded total k-excess. Our main result is as follows. Suppose k ≥ 2, 6 ≥ 0, and G is a connected graph satisfying the following condition: For every subset S of V(G), c(G\S) ≤ (k-2)\S\ + 2 + b. Then, G has a spanning tree with total k-excess at most 6.

Original language | English |
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Pages (from-to) | 289-295 |

Number of pages | 7 |

Journal | Ars Combinatoria |

Volume | 102 |

Publication status | Published - 2011 Oct 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Ars Combinatoria*,

*102*, 289-295.