MAX MIN Vertex Cover and the Size of Betti Tables


Let G be a finite simple graph on n vertices, that contains no isolated vertices, and let \(I(G) \subseteq S = K[x_1, \dots , x_n]\) be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of S/I(G). We show that if \({{\,\mathrm{pd}\,}}(S/I(G))\) attains its minimum possible value \(2\sqrt{n}-2\) then, with only one exception, \({{\,\mathrm{reg}\,}}(S/I(G)) = 1\). We also provide a full description of the spectrum of \({{\,\mathrm{pd}\,}}(S/I(G))\) when \({{\,\mathrm{reg}\,}}(S/I(G))\) attains its minimum possible value 1.

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The first author was supported by Louisiana BOR grant LEQSF(2017-19)-ENH-TR-25, and the second author was supported by JSPS KAKENHI 19H00637. The authors thank Martin Milanic for informing us that the inequality (1.1) was already known and drawing our attention to [3], and thank Antonio Macchia for pointing out to us that \(C_4\) should be included in our classification result. The authors also thank anonymous referees for a careful read and useful suggestions.

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Correspondence to Huy Tài Hà.

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Hà, H.T., Hibi, T. MAX MIN Vertex Cover and the Size of Betti Tables. Ann. Comb. (2021).

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  • Max min vertex cover
  • Min max independent set
  • Gap-free graph
  • Chordal graph
  • Regularity
  • Projective dimension
  • Betti table
  • Monomial ideal
  • Squarefree monomial
  • Edge ideal

Mathematics Subject Classification

  • 13D02
  • 05C70
  • 05E40