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Independent set sequence of some linear hypertrees

David Galvin, Courtney Sharpe

TL;DR

The paper extends the study of independent set sequences from graphs to linear hypertrees, proving that strong independent set sequences for linear hyperpaths and linear hyperstars are unimodal and in fact log-concave. It develops explicit formulas and recurrences for the uniform linear hyperpath counts $p^k_{n,\ell}$, including a combinatorial proof of the key summation formula, and analyzes the hypercomb family $C_{n,\ell}$ to obtain almost unimodality with a gap of at most $O(\sqrt{n})$. The results combine generating-function techniques, real-rootedness arguments (via claw-free graphs and Chudnovsky–Seymour), and a general almost-unimodality framework for recurrences, enabling partial unimodality results for a broad hypertree family. Together, these findings provide concrete evidence toward a hypertree analogue of the unimodality conjecture for trees, while outlining persistent open questions for more general hypertrees.

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 Erdős 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 $o(n)$.

Independent set sequence of some linear hypertrees

TL;DR

The paper extends the study of independent set sequences from graphs to linear hypertrees, proving that strong independent set sequences for linear hyperpaths and linear hyperstars are unimodal and in fact log-concave. It develops explicit formulas and recurrences for the uniform linear hyperpath counts , including a combinatorial proof of the key summation formula, and analyzes the hypercomb family to obtain almost unimodality with a gap of at most . The results combine generating-function techniques, real-rootedness arguments (via claw-free graphs and Chudnovsky–Seymour), and a general almost-unimodality framework for recurrences, enabling partial unimodality results for a broad hypertree family. Together, these findings provide concrete evidence toward a hypertree analogue of the unimodality conjecture for trees, while outlining persistent open questions for more general hypertrees.

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 Erdős 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 edges on the spine, and show that its strong independent set sequence is unimodal except possibly for a portion of length .
Paper Structure (8 sections, 6 theorems, 83 equations, 3 figures, 3 tables)

This paper contains 8 sections, 6 theorems, 83 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hypertrees
  3. Proofs
  4. Proof of Theorem \ref{['thm-hyperstar,hyperpath']}
  5. Deriving a formula for $p_{n,\ell}^k$
  6. Combinatorial proof of the formula for $p_{n,\ell}^k$
  7. Proofs of \ref{['fib-rec']} and \ref{['eq-Px-pure']}
  8. The independent set sequence of hypercombs and related hypergraphs

Key Result

Theorem 1.3

For $n \geq 1$ and $s_1, s_2, \ldots, s_n \geq 2$

Figures (3)

  • Figure 1: The first 5 edges of the linear hyperpaths $P_{n,5}$ (top) and $Q_{n,5}$ (bottom). Note that the first edge of $Q_{n,5}$ has one vertex fewer than the first edge of $P_{n,5}$.
  • Figure 2: The linear uniform hypercomb $C_{6,5}$ (top) and the auxiliary hypercomb $D_{6,5}$ (bottom).
  • Figure 3: An independent set of size $4$ in the hyperpath $P_{6,5}$. The marked edges ($e_1, e_2, e_3$ and $e_4$) are shaded in teal. The vertex $v_2$ is forced and the vertex $v_3$ is free.

Theorems & Definitions (8)

  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.7
  • Corollary 1.8
  • Claim 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3