Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Haruhide Matsuda; Kenta Ozeki; Tomoki Yamashita

doi:10.1007/s00373-012-1277-5

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Haruhide Matsuda, Kenta Ozeki, Tomoki Yamashita

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

13 Citations (Scopus)

Abstract

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n - 2. These conditions are best possible. A related conjecture also is proposed.

Original language	English
Pages (from-to)	429-437
Number of pages	9
Journal	Graphs and Combinatorics
Volume	30
Issue number	2
DOIs	https://doi.org/10.1007/s00373-012-1277-5
Publication status	Published - 2014 Mar

Keywords

Branch vertices
Claw-free graphs
Spanning trees

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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AU - Ozeki, Kenta

AU - Yamashita, Tomoki

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N2 - Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n - 2. These conditions are best possible. A related conjecture also is proposed.

AB - Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n - 2. These conditions are best possible. A related conjecture also is proposed.

KW - Branch vertices

KW - Claw-free graphs

KW - Spanning trees

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Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Abstract

Keywords

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