Abstract
Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all \(s \ge 1\), we obtain upper bounds for \({\text {reg}}(I(G)^s)\) for bipartite graphs. We then compare the properties of G and \(G'\), where \(G'\) is the graph associated with the polarization of the ideal \((I(G)^{s+1} : e_1\cdots e_s)\), where \(e_1,\cdots , e_s\) are edges of G. Using these results, we explicitly compute \({\text {reg}}(I(G)^s)\) for several subclasses of bipartite graphs.
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Acknowledgements
We would like to thank Adam Van Tuyl who pointed us to the article [5]. We would also like to thank Arindam Banerjee and Selvi Beyarslan for some useful discussions regarding the materials discussed in this paper. We heavily used the commutative algebra package, Macaulay 2, [13], for verifying whichever conjectures came to our mind. The third author is funded by National Board for Higher Mathematics, India.
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Jayanthan, A.V., Narayanan, N. & Selvaraja, S. Regularity of powers of bipartite graphs. J Algebr Comb 47, 17–38 (2018). https://doi.org/10.1007/s10801-017-0767-1
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DOI: https://doi.org/10.1007/s10801-017-0767-1