Spectral radius and Hamiltonicity of uniform hypergraphs

TL;DR

The paper addresses Hamiltonicity in $r$-uniform hypergraphs under spectral radius constraints, generalizing the Fiedler–Nikiforov graph result. It defines the spectral radius $\lambda(\mathcal{H})$ via the polynomial form and relates it to edge counts using the Bai–Lu bound, while leveraging Dirac-type hypergraph results of Kostochka–Luo–McCourt to prove sharp thresholds. The main contributions are: (i) a spectral-radius criterion $\lambda(\mathcal{H}) \ge \binom{n-2}{r-1}$ ensuring a Hamiltonian Berge path, and $> \binom{n-2}{r-1}$ ensuring a Hamiltonian Berge cycle, with explicit extremal structures $\mathcal{H} \cong K_{n-1}^r+v$ or $+e$; and (ii) a matching edge-count threshold, $|\mathcal{H}| \ge \binom{n-1}{r}$ (and $> \binom{n-1}{r}$), with the same near-complete obstructions. These results extend spectral-hamiltonicity from graphs to hypergraphs and connect hypergraph extremal theory with spectral methods. The findings are tight for $n \ge r+2$, and the proofs blend structural, combinatorial, and spectral techniques.

Abstract

Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $λ(\mathcal{H}) > \binom{n-2}{r-1}$ must contain a Hamiltonian Berge cycle unless $\mathcal{H}$ is the complete graph $K_{n-1}^r$ with one additional edge. This generalizes a result proved by Fiedler and Nikiforov for graphs. As part of our proof, we show that if $|\mathcal{H}| > \binom{n-1}{r}$, then $\mathcal{H}$ contains a Hamiltonian Berge cycle unless $\mathcal{H}$ is the complete graph $K_{n-1}^r$ with one additional edge, generalizing a classical theorem for graphs.

Theorems & Definitions (9)

• Theorem 1: Fiedler-Nikiforov
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Theorem 4: Kostochka-Luo-McCourt
• proof : Proof of Theorem \ref{['thm:edges']}
• Theorem 5: Bai-Lu
• proof : Proof of Theorem \ref{['thm:specrad']}