Spanning trees with few branch vertices
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A branch vertex in a tree is a vertex of degree at least three. We prove that for all positive integers s, every connected graph on n vertices with minimum degree at least (1/s+3+o(1))n contains a spanning tree having at most s branch vertices. Asymptotically, this is best possible and solves a problem of Flandrin, Kaiser, Kuz̆el, Li and Ryjác̆ek, which was originally motivated by an optimization problem in the design of optical networks.
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