On the maximum quartet distance between phylogenetic trees

TL;DR

This paper uses flag algebra techniques to improve bounds on the maximum quartet distance between phylogenetic trees, providing evidence supporting the conjecture that the maximum is at most two-thirds of all quartets.

Contribution

The authors improve the upper bound on the maximum quartet distance and provide evidence supporting the conjecture for specific tree types.

Findings

Maximum quartet distance is at most 0.69 times the total quartets.

Maximum distance between caterpillar trees is at most two-thirds of all quartets.

Supports the conjecture that the maximum is at most two-thirds.

Abstract

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on $n$ leaves is at most $(\frac 23 +o(1))\binom{n}{4}$. Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most $(0.69 +o(1))\binom{n}{4}$. We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most $(\frac 23 +o(1))\binom{n}{4}$.