### Abstract

A tree is called a k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. We prove the following theorem. Let k≥2 be an integer, and let G be a connected bipartite graph with bipartition (A,B) such that |A|≤|B|≤|A|+k-1. If ^{σ2}(G)≥(|G|-k+2)/2, then G has a spanning k-ended tree, where ^{σ2}(G) denotes the minimum degree sum of two non-adjacent vertices of G. Moreover, the condition on ^{σ2}(G) is sharp. It was shown by Las Vergnas, and Broersma and Tuinstra, independently that if a graph H satisfies ^{σ2}(H) ≥|H|-k+1 then H has a spanning k-ended tree. Thus our theorem shows that the condition becomes much weaker if a graph is bipartite.

Original language | English |
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Pages (from-to) | 2903-2907 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 313 |

Issue number | 24 |

DOIs | |

Publication status | Published - 2013 Oct 7 |

### Keywords

- Spanning k-ended tree
- Spanning tree
- Spanning tree with at most k leaves

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*313*(24), 2903-2907. https://doi.org/10.1016/j.disc.2013.09.002