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<p>When <i>I</i> is the edge ideal of a graph <i>G</i>, we use combinatorial properties, particularly property <i>P</i> on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on <i>R</i>/<i>I</i>(<i>G</i>). Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding <i>h</i>-vector can be calculated from a related graph using a simplified calculation on the <i>f</i>-vector, or independence vector, of the related graph. In the case when the graph is Cohen–Macaulay with a perfect matching of regular edges satisfying the separability criterion, the <i>h</i>-vector of <i>R</i>/<i>I</i>(<i>G</i>) will be precisely the <i>f</i>-vector of the Stanley–Reisner complex of a graph with half as many vertices as <i>G</i>.</p>

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Regular edges, matchings and Hilbert series

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月10日

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  • Joseph Brennan,
  • Susan Morey

摘要

When I is the edge ideal of a graph G, we use combinatorial properties, particularly property P on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on R/I(G). Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding h-vector can be calculated from a related graph using a simplified calculation on the f-vector, or independence vector, of the related graph. In the case when the graph is Cohen–Macaulay with a perfect matching of regular edges satisfying the separability criterion, the h-vector of R/I(G) will be precisely the f-vector of the Stanley–Reisner complex of a graph with half as many vertices as G.