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Regularity of powers of forests and cycles

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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.

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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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Correspondence to Huy Tài Hà.

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Beyarslan, S., Hà, H.T. & Trung, T.N. Regularity of powers of forests and cycles. J Algebr Comb 42, 1077–1095 (2015).

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