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Componentwise linearity of powers of cover ideals

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Abstract

Let G be a finite simple graph and J(G) denote its vertex cover ideal in a polynomial ring over a field. The k-th symbolic power of J(G) is denoted by \(J(G)^{(k)}\). In this paper, we give a criterion for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that \(J(G)^{(k)}\) is a componentwise linear ideal for some (equivalently, for all) \(k \ge 2\) when G is a graph such that \(G {\setminus } N_G[A]\) has a simplicial vertex for any independent set A of G. Using this result, we prove that \(J(G)^{(k)}\) is a componentwise linear ideal for several classes of graphs for all \(k \ge 2\). In particular, if G is a bipartite graph, then J(G) is a componentwise linear ideal if and only if \(J(G)^k\) is a componentwise linear ideal for some (equivalently, for all) \(k \ge 2\).

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Notes

  1. The W-graph is named in honor of Russ Woodroofe.

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Acknowledgements

The authors would like to thank Huy Tài Hà for valuable discussions. The authors extensively used Macaulay2, [16], and the packages EdgeIdeals, [13], SimplicialDecomposability, [4], SymbolicPowers, [8], for testing their computations. The first author is supported by DST, Govt of India under the DST-INSPIRE [DST/Inspire/04/2019/001353] Faculty Scheme.

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Authors and Affiliations

  1. Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai, 603103, India

    S. Selvaraja

  2. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Joseph W. Skelton

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  1. S. Selvaraja
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Correspondence to Joseph W. Skelton.

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Dedicated to Professor Jürgen Herzog on the occasion of his 80th birthday.

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Selvaraja, S., Skelton, J.W. Componentwise linearity of powers of cover ideals. J Algebr Comb 57, 111–134 (2023). https://doi.org/10.1007/s10801-022-01160-z

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