The even-uniform hypergraph Moore bound

2026-07-15

AI Summary

This work investigates the relationship between the minimum size of nontrivial even covers and the average edge density in $k$-uniform hypergraphs, aiming to verify Feige's conjectured hypergraph Moore bound. By constructing colored walks in the Kikuchi graph derived from the hypergraph and employing polynomial interpolation techniques to precisely control the growth of combinatorial structures, we present a concise and rigorous proof. Our approach fully confirms Feige’s original conjecture for all even $k \geq 4$ without any extraneous polylogarithmic factors, thereby eliminating the error terms present in prior results and significantly streamlining the proof framework. This establishes an exact bound linking even cover size and hyperedge density.

Abstract

The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a $k$-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all even $k\ge 4$, with no superfluous polylogarithmic factors. Our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using a polynomial interpolation method.

Problem

• hypergraph
• Moore bound
• even cover
• girth
• uniform hypergraph

Innovation

• hypergraph Moore bound
• even cover
• Kikuchi graph
• polynomial interpolation
• uniform hypergraph