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A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution

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Abstract

Let I(G) be the edge ideal of a simple graph G. We prove that \(I(G)^2\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 3\). Similarly, we show that \(I(G)^3\) has a linear free resolution if and only if G is gap-free and \({{\,\textrm{reg}\,}}I(G) \le 4\). We deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs \({{\,\textrm{reg}\,}}(I(G)^s) = \max ({{\,\textrm{reg}\,}}I(G) + s-1,2s)\), for \(s =2,3\).

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Acknowledgements

Nguyen Cong Minh is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 101.04\(-\)2021.19. Finally, we are grateful to Prof. Seyed Fakhari and the anonymous referees for their thoughtful suggestions and comments to improve the readability of our manuscript.

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Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam

    Nguyen Cong Minh

  2. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Thanh Vu

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  1. Nguyen Cong Minh
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  2. Thanh Vu
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Correspondence to Thanh Vu.

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Dedicated to Professor Le Tuan Hoa on the occasion of his 65th birthday.

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Minh, N.C., Vu, T. A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution. Combinatorica 44, 337–353 (2024). https://doi.org/10.1007/s00493-023-00074-z

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