Degree conditions and degree bounded trees

Haruhide Matsuda; Hajime Matsumura

doi:10.1016/j.disc.2007.12.099

Degree conditions and degree bounded trees

Haruhide Matsuda, Hajime Matsumura

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σ_k (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σ_k (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σ_{k - w (G - A)} (A) ≥ | A | - 1, then G contains a spanning tree T such that d_T (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

Original language	English
Pages (from-to)	3653-3658
Number of pages	6
Journal	Discrete Mathematics
Volume	309
Issue number	11
DOIs	https://doi.org/10.1016/j.disc.2007.12.099
Publication status	Published - 2009 Jun 6

Keywords

Degree bounded tree
Degree sum condition
Spanning tree
Tree

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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title = "Degree conditions and degree bounded trees",

abstract = "We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σk (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σk - w (G - A) (A) ≥ | A | - 1, then G contains a spanning tree T such that dT (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Ger{\"u}sten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.",

keywords = "Degree bounded tree, Degree sum condition, Spanning tree, Tree",

author = "Haruhide Matsuda and Hajime Matsumura",

note = "Funding Information: The first author{\textquoteright}s research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005. The second author{\textquoteright}s research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.",

year = "2009",

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doi = "10.1016/j.disc.2007.12.099",

language = "English",

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AU - Matsuda, Haruhide

AU - Matsumura, Hajime

N1 - Funding Information: The first author’s research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005. The second author’s research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.

PY - 2009/6/6

Y1 - 2009/6/6

N2 - We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σk (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σk - w (G - A) (A) ≥ | A | - 1, then G contains a spanning tree T such that dT (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

AB - We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk (A) the minimum value of the degree sum in G of any k independent vertices in A and by w (G - A) the number of components in the induced subgraph G - A. Our main results are the following: (i) If σk (A) ≥ | V (G) | - 1, then G contains a tree T with maximum degree at most k and A ⊆ V (T). (ii) If σk - w (G - A) (A) ≥ | A | - 1, then G contains a spanning tree T such that dT (x) ≤ k for every x ∈ A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp.

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