Asymptotically Enumerating Independent Sets in Regular $k$-Partite $k$-Uniform Hypergraphs

TL;DR

This work develops a cluster-expansion framework to asymptotically enumerate independent sets in regular $k$-partite $k$-uniform hypergraphs by expressing the count as a local, polymer-model partition function. Under Reg$(t)$, Exp$_1(\alpha_{k,t})$, Exp$_2(\beta)$, and Def$(b(n))$, the paper proves that $|\mathcal I(G)|$ is captured up to $(1+\rho)$ by a locally computable exponential involving clusters of size at most $t$, with a special closed form for linear hypergraphs when $t=1$. The main technical contribution is verifying the Kotecký-Preiss condition for the polymer model and carefully bounding contributions from larger clusters through expansion properties and 2-linkedness, enabling truncation of the cluster expansion. The results extend efficient, local-structure-based enumeration techniques from bipartite graphs to hypergraphs and open avenues for considering weighted counts and homomorphism-type problems in regular hypergraph settings.

Abstract

The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been extensively used for graphs, surprisingly little is known about analogous questions in the context of hypergraphs. In this work, we apply this method to asymptotically determine the number of independent sets in regular $k$-partite $k$-uniform hypergraphs which satisfy natural expansion properties. The resulting formula depends only on the local structure of the hypergraph, making it computationally efficient. In particular, we provide a simple closed-form expression for linear hypergraphs.

Theorems & Definitions (36)

• Definition 1.1
• Theorem 1.2
• Theorem 2.1
• proof : Proof of \ref{['thm: t=1']}
• Lemma 2.2
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2