Spanning Trees: A Survey

Spanning k-ended trees of 3-regular connected graphs

“…After [1], many researchers have defined other closure concepts for various graph properties. More on k-ended tree and spanning tree can be found in [6,7,8,9]. In this paper, we obtain sufficient conditions for spanning k-ended trees of 3-regular connected graphs and with construction sequence of graphs like G m , we will show this condition is sharp.…”

Spanning trees with at most 4 leaves in K1,5-free graphs

“…There are several well-known conditions (such as the independence number conditions and the degree sum conditions) ensuring that a graph G contains a spanning tree with a bounded number of leaves or branch vertices (see the survey paper [12] and the references cited therein for details). Win [13] obtained a sufficient condition related to the independence number for k-connected graphs, which confirms a conjecture of Las Vergnas [9].…”

Spanning Trees with Few Branch Vertices

“…Motivated by this, Gargano et al [15] conjectured that for all s ≥ 1, if G is a connected graph on n vertices with δ(G) ≥ (n − 1)/(s + 2), then G contains a spanning tree with at most s branch vertices. Later, Flandrin, Kaiser, Kužel, Li and Ryjáček [12,Problem 11] asked if the much stronger bound of δ(G) ≥ n/(s + 3) + C is sufficient and then Ozeki and Yamashita [21,Conjecture 30] conjectured a precise value for the constant term 1 . Note that even the approximate version of the conjecture by Flandrin et al has not been verified for any s ≥ 1 and the original (weaker) conjecture of Gargano et al has not been verified for any s ≥ 2.…”