Spanning Trees Whose Stems are Spiders

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.