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<p>Recently, Zhang and Wu proved a conjecture of Kalai and Meshulam, showing that for every graph <i>G</i> without induced cycles of length divisible by 3, the sum of all reduced Betti numbers of its independence complex <i>I</i>(<i>G</i>) is at most 1. We extend this result to the hypergraph setting. Namely, we show that the same conclusion holds for any hypergraph <i>H</i> that does not contain a Berge cycle of length divisible by 3. This establishes a broader connection between forbidden cycle structures and the topological simplicity of independence complexes. As a key tool, we introduce a hypergraph analogue of Barmak’s star cluster theorem for graphs. This new theorem implies, in particular, that if a hypergraph <i>H</i> has a vertex <i>v</i> that is not isolated and is not contained in an induced Berge cycle of length 3, then there exists a hypergraph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> with fewer vertices than <i>H</i> such that the independence complex of <i>H</i> is homotopy equivalent to the suspension of the independence complex of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>.</p>

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Topology of Independence Complexes and Cycle Structure of Hypergraphs

  • 期刊论文

  • 发表日期：

    2026年1月28日

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  • Jinha Kim

摘要

Recently, Zhang and Wu proved a conjecture of Kalai and Meshulam, showing that for every graph G without induced cycles of length divisible by 3, the sum of all reduced Betti numbers of its independence complex I(G) is at most 1. We extend this result to the hypergraph setting. Namely, we show that the same conclusion holds for any hypergraph H that does not contain a Berge cycle of length divisible by 3. This establishes a broader connection between forbidden cycle structures and the topological simplicity of independence complexes. As a key tool, we introduce a hypergraph analogue of Barmak’s star cluster theorem for graphs. This new theorem implies, in particular, that if a hypergraph H has a vertex v that is not isolated and is not contained in an induced Berge cycle of length 3, then there exists a hypergraph \(H'\) H ′ with fewer vertices than H such that the independence complex of H is homotopy equivalent to the suspension of the independence complex of \(H'\) H ′ .