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.
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Matsuda, H., Ozeki, K. & Yamashita, T. Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph. Graphs and Combinatorics 30, 429–437 (2014). https://doi.org/10.1007/s00373-012-1277-5
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DOI: https://doi.org/10.1007/s00373-012-1277-5