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arXiv · 2501.18540

Leaf-to-leaf paths of many lengths

Francesco Di Braccio·Kyriakos Katsamaktsis·Alexandru Malekshahian
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Abstract

We prove that every tree of maximum degree $Δ$ with $\ell$ leaves contains paths between leaves of at least $\log_{Δ-1}((Δ-2)\ell)$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and Szabó. We also make progress towards another conjecture of the same authors, by proving that every tree with no vertex of degree 2 and diameter at least $N$ contains $N^{2/3}/6$ distinct leaf-to-leaf path lengths between $0$ and $N$.

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Francesco Di Braccio, Kyriakos Katsamaktsis, Alexandru Malekshahian. 2025-04-17. Leaf-to-leaf paths of many lengths. https://arxiv.org/abs/2501.18540

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