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 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 | |
| Publication status | Published - 2009 Jun 6 |
| Externally published | Yes |
Keywords
- Degree bounded tree
- Degree sum condition
- Spanning tree
- Tree
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
Cite this
Degree conditions and degree bounded trees. / Matsuda, Haruhide; Matsumura, Hajime.
In: Discrete Mathematics, Vol. 309, No. 11, 06.06.2009, p. 3653-3658.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Degree conditions and degree bounded trees
AU - Matsuda, Haruhide
AU - Matsumura, Hajime
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.
KW - Degree bounded tree
KW - Degree sum condition
KW - Spanning tree
KW - Tree
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U2 - 10.1016/j.disc.2007.12.099
DO - 10.1016/j.disc.2007.12.099
M3 - Article
VL - 309
SP - 3653
EP - 3658
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 11
ER -