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arXiv · 2303.07710

A note on the flip distance between non-crossing spanning trees

Nicolas Bousquet·Valentin Gledel·Jonathan Narboni·Théo Pierron
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Abstract

We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive $2n-\Omega(1)$ upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a $2n-\Omega(\log n)$ bound. We improve their result with a $2n-\Omega(\sqrt{n})$ upper bound, and we strengthen and shorten the proofs of several of their results.

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Nicolas Bousquet, Valentin Gledel, Jonathan Narboni, Théo Pierron. 2023-03-14. A note on the flip distance between non-crossing spanning trees. https://arxiv.org/abs/2303.07710

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