TY - JOUR

T1 - Degree Sum Condition for the Existence of Spanning k-Trees in Star-Free Graphs

AU - Furuya, Michitaka

AU - Maezawa, Shun Ichi

AU - Matsubara, Ryota

AU - Matsuda, Haruhide

AU - Tsuchiya, Shoichi

AU - Yashima, Takamasa

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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.

AB - 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.

KW - degree sum condition

KW - k-tree

KW - spanning tree

KW - star-free

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U2 - 10.7151/dmgt.2234

DO - 10.7151/dmgt.2234

M3 - Article

AN - SCOPUS:85079613544

JO - Discussiones Mathematicae - Graph Theory

JF - Discussiones Mathematicae - Graph Theory

SN - 1234-3099

ER -