An AI enhanced approach to the tree unimodality conjecture
This work addresses a long-standing conjecture in graph theory, providing extensive evidence against it, which is significant for mathematicians and computer scientists studying combinatorial properties.
The paper tackled the log-concavity conjecture for independence sequences of trees by using the AI architecture PatternBoost to find counter-examples, resulting in the discovery of tens of thousands of new counter-examples with vertex set sizes ranging from 27 to 101.
Given a graph $G$, its independence sequence is the integral sequence $a_1,a_2,...,a_n$, where $a_i$ is the number of independent sets of vertices of size i. In the late 80's Alavi, Erdos, Malde, Schwenk showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever $G$ is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that $a_i^2$ is always above $a_{i-1}a_{i+1}$. This conjecture stood for many years, until in 2023, Kadrawi, Levit, Yosef, and Mizrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. In this paper, we use the AI architecture PatternBoost, developed by Charton, Ellenberg, Wagner, and Williamson to train a machine to find counter-examples to the log-concavity conjecture. We will discuss the successes of this approach - finding tens of thousands of new counter-examples to log-concavity with vertex set sizes varying from 27 to 101 - and some of its fascinating failures.