Independent set sequence of some linear hypertrees

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 $o(n)$.

💡 Research Summary

The paper investigates the independent‑set sequence of hypergraphs, focusing on linear hypertrees, which are hypergraphs whose edges intersect in at most one vertex and whose underlying incidence structure forms a tree. The authors restrict attention to strong independent sets—sets containing at most one vertex from each hyperedge—because for ordinary graphs the notions of weak and strong independence coincide. Their goal is to determine whether the strong independent‑set sequence (the numbers i_k of strong independent sets of size k) is always unimodal for linear hypertrees, extending the long‑standing conjecture for ordinary trees.
The first main result (Theorem 1.3) treats two fundamental families of linear hypertrees: linear hyperpaths P(s₁,…,sₙ) and linear hyperstars S(s₁,…,sₙ). Using an interlacing argument based on results of Chudnovsky–Seymour and Hamidoune, the authors prove that the independent‑set sequence of any linear hyperpath is real‑rooted; consequently it is log‑concave and therefore unimodal. For linear hyperstars they give a direct proof of log‑concavity, again implying unimodality. This establishes that the two extremal linear hypertrees (maximal and minimal diameter) have unimodal sequences, supporting the broader conjecture.
The paper then turns to the uniform linear hyperpath Pₙ,ℓ (all edges have size ℓ≥2). Let p_{k,n,ℓ} denote the number of strong independent sets of size k. By analysing the generating polynomial P_{n,ℓ}(x)=∑{k}p{k,n,ℓ}x^{k}, they derive a two‑term recurrence
 P_{n,ℓ}(x) = (1+(ℓ−2)x)P_{n−1,ℓ}(x) + x P_{n−2,ℓ}(x) for n≥3,
with initial conditions P_{0,ℓ}(x)=1 and P_{1,ℓ}(x)=1+ℓx. Translating coefficients yields the combinatorial recurrence
 p_{k,n,ℓ}=p_{k,n−1,ℓ}+p_{k−1,n−2,ℓ}+(ℓ−2)p_{k−1,n−1,ℓ} (k≥2, n≥2).
From these recurrences the authors conjecture—and then prove by induction—a closed formula:
 p_{k,n,ℓ}= (ℓ−1)^{2k−2} ∑{j=0}^{ℓ−2} \binom{k−2j}{j},\binom{n−j}{k}.
When ℓ=2 this reduces to the well‑known binomial expression (\binom{n−k+2}{k}) for ordinary paths. The paper supplies two complementary explanations: an algebraic derivation via generating functions, and a combinatorial argument that partitions independent sets according to the first edge that contributes a vertex, thereby justifying the double‑sum structure.
Having settled the uniform hyperpath, the authors study a more intricate family: the uniform linear hypercomb C{n,ℓ}. This structure is obtained from P_{n,ℓ} by attaching, for each internal vertex, an extra ℓ‑edge that contains the vertex and ℓ−1 new pendant vertices, and also adding two pendant ℓ‑edges at the two ends. For ℓ=2 this is the classic centipede graph, known to have a real‑rooted independent‑set polynomial. The authors prove (Theorem 1.7) that the independence polynomial C_{n,ℓ}(x) satisfies a three‑term recurrence:
 C_{n,ℓ}(x) = (1+(ℓ−1)x)