An AI enhanced approach to the tree unimodality conjecture

TL;DR

This paper employs an AI-based method to identify counter-examples to the log-concavity conjecture of independence sequences in trees, discovering tens of thousands of new counter-examples for trees with 27 to 101 vertices.

Contribution

It introduces an AI-enhanced approach using PatternBoost to find counter-examples to a longstanding conjecture about tree independence sequences, significantly expanding known cases.

Findings

Discovered tens of thousands of new counter-examples

Found counter-examples in trees with 27 to 101 vertices

Demonstrated successes and limitations of AI in combinatorial conjecture testing

Abstract

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.