### 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 |
---|---|

Pages (from-to) | 2903-2907 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 313 |

Issue number | 24 |

DOIs | |

Publication status | Published - 2013 |

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

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

**Spanning k-ended trees of bipartite graphs.** / Kano, Mikio; Matsuda, Haruhide; Tsugaki, Masao; Yan, Guiying.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 313, no. 24, pp. 2903-2907. https://doi.org/10.1016/j.disc.2013.09.002

}

TY - JOUR

T1 - Spanning k-ended trees of bipartite graphs

AU - Kano, Mikio

AU - Matsuda, Haruhide

AU - Tsugaki, Masao

AU - Yan, Guiying

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Spanning k-ended tree

KW - Spanning tree

KW - Spanning tree with at most k leaves

UR - http://www.scopus.com/inward/record.url?scp=84884873963&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884873963&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2013.09.002

DO - 10.1016/j.disc.2013.09.002

M3 - Article

VL - 313

SP - 2903

EP - 2907

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 24

ER -