### Abstract

For an integer k ≥ 2, a k-tree T is defined as a tree with maximum degree at most k. If a k-tree T spans a graph G, then T is called a spanning k-tree of G. Since a spanning 2-tree is a Hamiltonian path, a spanning k-tree is an extended concept of a Hamiltonian path. The first result, implying the existence of k-trees in star-free graphs, was by Caro, Krasikov, and Roditty in 1985, and independently, Jackson and Wormald in 1990, who proved that for any integer k with k ≥ 3, every connected K_{1,k}-free graph contains a spanning k-tree. In this paper, we focus on a sharp condition that guarantees the existence of a spanning k-tree in K_{1,k+1}-free graphs. In particular, we show that every connected K_{1,k+1}-free graph G has a spanning k-tree if the degree sum of any 3k - 3 independent vertices in G is at least |G| - 2, where |G| is the order of G.

Original language | English |
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Journal | Discussiones Mathematicae - Graph Theory |

DOIs | |

Publication status | Accepted/In press - 2020 Jan 1 |

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### Keywords

- degree sum condition
- k-tree
- spanning tree
- star-free

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discussiones Mathematicae - Graph Theory*. https://doi.org/10.7151/dmgt.2234