TITLE:
Closure for Spanning Trees with k-Ended Stems
AUTHORS:
Zheng Yan
KEYWORDS:
Closure, Spanning Tree, Stem, k-End Stem
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.4 No.3,
June
26,
2014
ABSTRACT: Let T be a tree. The set of leaves of Τ is denoted by Leaf(Τ). The subtree Τ—Leaf(Τ) of T is called the stem of Τ. A stem is called a k-ended stem if it has at most k-leaves in it. In this paper, we prove
the following theorem. Let G be a connected graph and k≥2 be an integer. Let u and ν be a pair of nonadjacent vertices in G. Suppose that |NG(u)∪NG(v)|≥|G|-k-1. Then G has a spanning tree with k-ended stem if and only if G+uv has a spanning tree with k-ended stem. Moreover, the condition on |NG(u)∪NG(v)| is sharp.