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Regularity of symbolic powers of cover ideals of graphs

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Abstract

Let G be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw–free graphs. Assume that J(G) is the cover ideal G and \(J(G)^{(k)}\) is its k-th symbolic power. We prove that

$$\begin{aligned} k\mathrm{deg}(J(G))\le \mathrm{reg}(J(G)^{(k)})\le (k-1)\mathrm{deg}(J(G))+|V(G)|-1. \end{aligned}$$

We also determine families of graphs for which the above inequalities are equality.

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Acknowledgements

The author thanks the referee for careful reading of the paper and for useful comments.

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  1. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran

    S. A. Seyed Fakhari

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  1. S. A. Seyed Fakhari
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Seyed Fakhari, S.A. Regularity of symbolic powers of cover ideals of graphs. Collect. Math. 70, 187–195 (2019). https://doi.org/10.1007/s13348-018-0228-8

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  • DOI: https://doi.org/10.1007/s13348-018-0228-8

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