Abstract
Let \(\tilde {C_{n}}\) be the graph by adding an ear to C n and \(I=I(\tilde {C_{n}})\) be its edge ideal. In this paper, we prove that \(\operatorname {reg}(I^{s})=2s+\lfloor \frac {n+1}{3}\rfloor -1\) for all s ≥ 1. Let G be the bicyclic graph C m ⊔ C n with edge ideal I = I(G); we compute the regularity of I s for all s ≥ 1. In particular, in some cases, we get \(\operatorname {reg}(I^{s})=2s+\lfloor \frac {m}{3}\rfloor +\lfloor \frac {n}{3}\rfloor -1\) for all s ≥ 2.
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Acknowledgments
The author would like to thank Professor Zhongming Tang for his helpful discussion and S. A. Seyed Fakhari for his valuable suggestions. The author is also grateful to the referee for his/her nice comments.
This work was supported by the National Natural Science Foundation of China (11501397), the Natural Science Foundation of Jiangsu Province (BK20140300), and the Jiangsu Government Scholarship for Overseas Studies.
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Gu, Y. Regularity of Powers of Edge Ideals of Some Graphs. Acta Math Vietnam 42, 445–454 (2017). https://doi.org/10.1007/s40306-017-0204-5
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DOI: https://doi.org/10.1007/s40306-017-0204-5