TY - JOUR

T1 - Degree conditions and degree bounded trees

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.

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

AN - SCOPUS:67349259976

SN - 0012-365X

VL - 309

SP - 3653

EP - 3658

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 11

ER -