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<p>In this paper, we study the componentwise linearity of symbolic powers of edge ideals. We propose the conjecture that all symbolic powers of the edge ideal of a cochordal graph are componentwise linear. This conjecture is verified for some families of cochordal graphs, including complements of block graphs and complements of proper interval graphs. As a corollary, Minh’s conjecture is established for such families. Moreover, we show that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(I(G)^{(2)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is componentwise linear, for any cochordal graph <i>G</i>.</p>

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Componentwise linear symbolic powers of edge ideals and Minh’s conjecture

  • 期刊论文

  • 发表日期：

    2025年12月7日

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  • Antonino Ficarra,
  • Somayeh Moradi,
  • Tim Römer

摘要

In this paper, we study the componentwise linearity of symbolic powers of edge ideals. We propose the conjecture that all symbolic powers of the edge ideal of a cochordal graph are componentwise linear. This conjecture is verified for some families of cochordal graphs, including complements of block graphs and complements of proper interval graphs. As a corollary, Minh’s conjecture is established for such families. Moreover, we show that \(I(G)^{(2)}\) I ( G ) ( 2 ) is componentwise linear, for any cochordal graph G.