Conditions for Implicit-Degree Sum for Spanning Trees with Few Leaves in K_1,4-Free Graphs

A graph with n vertices is called an n-graph. A spanning tree with at most k leaves is referred to as a spanning k-ended tree. Spanning k-ended trees are important in various fields such as network design, graph theory, and communication networks. They provide a structured way to connect all the nodes in a network while ensuring efficient communication and minimizing unnecessary connections. In addition, they serve as fundamental components for algorithms in routing, broadcasting, and spanning tree protocols. However, determining whether a connected graph has a spanning k-ended tree or not is NP-complete. Therefore, it is important to identify sufficient conditions for the existence of such trees. The implicit-degree proposed by Zhu, Li, and Deng is an important indicator for the Hamiltonian problem and the spanning k-ended tree problem. In this article, we provide two sufficient conditions for K-1,K-4-free connected graphs to have spanning k-ended trees for k = 2, 3. We prove the following: Let G be a K-1,K-4-free connected n-graph. For k = 2, 3, if the implicit-degree sum of any k + 1 independent vertices of G is at least n - k + 2, then G has a spanning k-ended tree. Moreover, we give two examples to show that the lower bounds n and n - 1 are the best possible.