Geodesic
Degree Sum Condition for the Existence of Spanning k-Trees in Star-Free Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 5-13

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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 K1,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 K1,k+1-free graphs. In particular, we show that every connected K1,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.
Keywords: spanning tree, k -tree, star-free, degree sum condition 
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     author = {Furuya, Michitaka and Maezawa, Shun-ichi and Matsubara, Ryota and Matsuda, Haruhide and Tsuchiya, Shoichi and Yashima, Takamasa},
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Furuya, Michitaka; Maezawa, Shun-ichi; Matsubara, Ryota; Matsuda, Haruhide; Tsuchiya, Shoichi; Yashima, Takamasa. Degree Sum Condition for the Existence of Spanning k-Trees in Star-Free Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 5-13. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a0/