Leaf-to-leaf paths of many lengths

TL;DR

This paper proves a lower bound on the number of distinct leaf-to-leaf path lengths in trees with bounded degree, settling a conjecture and advancing another related conjecture in graph theory.

Contribution

It establishes a strong lower bound on leaf-to-leaf path lengths in trees, confirming a conjecture and making progress on another related conjecture.

Findings

Lower bound of 1og_{\u03b4-1}((4-2)3) on distinct leaf-to-leaf path lengths

Proof that trees with no degree-2 vertices and large diameter have N^{2/3}/6 distinct leaf-to-leaf path lengths

Resolution of a conjecture by Narins, Pokrovskiy, and Szabf3

Abstract

We prove that every tree of maximum degree $\Delta$ with $\ell$ leaves contains paths between leaves of at least $\log_{\Delta-1}((\Delta-2)\ell)$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and Szab\'o. 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 $N$.