A Spectral Proof of the Hypergraph Moore Bound

Abstract:A nonempty subfamily of a $k$-uniform hypergraph is an \emph{even cover} if every vertex lies in an even number of its hyperedges; for $k=2$ these are edge-disjoint unions of cycles, so the minimum size of an even cover is the natural hypergraph analogue of girth. We prove Feige's 2008 conjecture on the hypergraph Moore bound: there are absolute constants $A$ and $C$ (independent of $k$) such that for every $k\ge3$ and every $1\le\ell\le n$, any $k$-uniform hypergraph on $n$ vertices with more than $C\,n^{k/2}/\ell^{k/2-1}$ hyperedges contains an even cover of size at most $A\,\ell\log(en/\ell)$. Our proof is based on sharp spectral bounds for Kikuchi matrices, which we expect to be of independent interest; we apply them to the refutation of random constraint satisfaction problems in a companion paper.

Combinatorics (math.CO); Discrete Mathematics (cs.DM); Quantum Physics (quant-ph)

arXiv:2607.26028 [math.CO]

(or arXiv:2607.26028v1 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2607.26028 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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