TY - JOUR

T1 - Spanning k-ended trees of bipartite graphs

AU - Kano, Mikio

AU - Matsuda, Haruhide

AU - Tsugaki, Masao

AU - Yan, Guiying

N1 - Funding Information:
The first author was partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) . The third author was supported by Chinese Academy of Science Fellowship for Young International Scientists (Grant No. 2012Y1JA0004 ).

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

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U2 - 10.1016/j.disc.2013.09.002

DO - 10.1016/j.disc.2013.09.002

M3 - Article

AN - SCOPUS:84884873963

VL - 313

SP - 2903

EP - 2907

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 24

ER -