Abstract
Let T be a tree. The set of leaves of T is denoted by Leaf(T). The subtree \(T-Leaf(T)\) is called the stem of T. A spider is a tree having at most one vertex with degree greater than two. In Gargano et al. (Discrete Math 285:83–95, 2004), it is shown that if a connected graph G satisfies \(\delta (G)\ge (|G|-1)/3\), then G has a spanning spider. In this paper, we prove that if \(\sigma _4^4(G)\ge |G|-5\), then G has a spanning tree whose stem is a spider, where \(\sigma _4^4(G)\) denotes the minimum degree sum of four vertices of G such that the distance between any two of their vertices are at least four. Moreover, we show that this condition is sharp.
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The first author was supported by JSPS KAKENHI Grant Number 25400187. The second author was supported by Doctoral Fund of Yangtze University Grant Number 80107001 and The Yangtze Youth Fund. This work was done while the second author was in Ibaraki University.
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Kano, M., Yan, Z. Spanning Trees Whose Stems are Spiders. Graphs and Combinatorics 31, 1883–1887 (2015). https://doi.org/10.1007/s00373-015-1618-2
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DOI: https://doi.org/10.1007/s00373-015-1618-2