Independent set sequence of some linear hypertrees

TL;DR

This paper investigates the properties of independent set sequences in linear hypertrees, establishing unimodality and log-concavity for specific classes like hyperpaths and hyperstars, and providing explicit formulas for uniform cases.

Contribution

It extends the study of independent set sequences from trees to hypertrees, proving unimodality and log-concavity for certain linear hypergraphs and deriving explicit formulas for uniform linear hyperpaths.

Findings

Strong independent set sequences of linear hyperpaths are unimodal and log-concave.

Strong independent set sequences of linear hyperstars are unimodal and log-concave.

Explicit formulas for the number of strong independent sets in uniform linear hyperpaths.

Abstract

The independent set sequence of trees has been well studied, with much effort devoted to the (still open) question of Alavi, Malde, Schwenk and Erd\H{o}s on whether the independent set sequence of a tree is always unimodal. Much less attention has been given to the independent set sequence of hypertrees. Here we study some natural first questions in this realm. We show that the strong independent set sequences of linear hyperpaths and of linear hyperstars are unimodal (actually, log-concave). For uniform linear hyperpaths we obtain explicit expressions for the number of strong independent sets of each possible size, both via generating functions and via combinatorial arguments. We also consider the uniform linear hypercomb with $n$ edges on the spine, and show that its strong independent set sequence is unimodal except possibly for a portion of length $o(n)$.