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Journal of Algebraic Combinatorics

, Volume 42, Issue 4, pp 1077–1095 | Cite as

Regularity of powers of forests and cycles

  • Selvi Beyarslan
  • Huy Tài Hà
  • Trân Nam Trung
Article

Abstract

Let G be a graph and let \(I = I(G)\) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of \(I^s\) for all \(s \ge 1\). In particular, for these classes of graphs, we provide the asymptotic linear function \({{\mathrm{reg}}}(I^s)\) as \(s \gg 0\), and the initial value of s starting from which \({{\mathrm{reg}}}(I^s)\) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.

Keywords

Regularity Powers of ideal Edge ideal Monomial ideal  Asymptotic linearity of regularity 

Notes

Acknowledgments

Part of this work was done while Hà and Trung were at the Vietnam Institute of Advanced Studies in Mathematics (VIASM) in Hanoi, Vietnam. We would like to thank VIASM for its hospitality. Hà is partially supported by the Simons Foundation (Grant #279786). We would also like to thank an anonymous referee for many helpful comments.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Selvi Beyarslan
    • 1
  • Huy Tài Hà
    • 1
  • Trân Nam Trung
    • 2
  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.Institute of MathematicsVietnam Academy of Science and Technology (VAST)HanoiVietnam

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