Trees with non log-concave independent set sequences

TL;DR

This paper constructs specific trees with large independence numbers where the independent set sequence fails to be log-concave, resolving a conjecture about the behavior of such sequences in trees.

Contribution

It provides a family of trees with large independence numbers that demonstrate the failure of log-concavity in their independent set sequences, answering a longstanding conjecture.

Findings

Identified trees with large independence numbers where log-concavity fails

Demonstrated the failure occurs near a specific fraction of the independence number

Resolved a conjecture by Kadrawi and Levit regarding independent set sequences

Abstract

We construct a family of trees with independence numbers going to infinity for which the log-concavity relation for the independent set sequence of a tree $T$ in the family fails at around $\alpha(T)\left(1-1/(16\log \alpha(T))\right)$. Here $\alpha(T)$ is the independence number of $T$. This resolves a conjecture of Kadrawi and Levit.