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Conditions for Implicit-Degree Sum for Spanning Trees with Few Leaves in K 1,4 -Free Graphs.
- Source :
- Mathematics (2227-7390); Dec2023, Vol. 11 Issue 24, p4981, 13p
- Publication Year :
- 2023
-
Abstract
- 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<subscript>1,4</subscript>-free connected graphs to have spanning k-ended trees for k = 2, 3. We prove the following: Let G be a K<subscript>1,4</subscript>-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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 11
- Issue :
- 24
- Database :
- Complementary Index
- Journal :
- Mathematics (2227-7390)
- Publication Type :
- Academic Journal
- Accession number :
- 174461131
- Full Text :
- https://doi.org/10.3390/math11244981