Trees with non log-concave independent set sequences

TL;DR

The paper addresses whether the independent set sequence $(i_t(T))_{t=0}^{\alpha(T)}$ of a tree is always log-concave, demonstrating that this property can fail even as $\alpha(T)$ grows without bound. It introduces the two-parameter family $T_{m,t}$, shows that for large $t$ and $m$ with $t \le m \le 2^{t/16}$ the log-concavity breaks at $mt+2$, and derives asymptotic counts $i_{T_{m,t}}(mt+2)$ and $i_{T_{m,t}}(mt+3)$ to certify the inequality. This resolves Kadrawi and Levit's conjecture and clarifies limitations of log-concavity in trees, while further discussing structural reasons and open questions about how far the breakdown can occur. Additionally, it verifies that the auxiliary graph family $S_{t,2}$ remains log-concave, contributing a piece to the overall understanding of independent set sequences in tree-like structures.

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 $α(T)\left(1-1/(16\log α(T))\right)$. Here $α(T)$ is the independence number of $T$. This resolves a conjecture of Kadrawi and Levit.

Figures (1)

• Figure 1: The tree $T_{4,3}$. The root is the topmost vertex.

Theorems & Definitions (6)

• Conjecture 1.2
• Theorem 1.3
• Theorem 2.1
• proof
• Lemma 3.3
• proof