On the regularity of small symbolic powers of edge ideals of graphs
DOI:
https://doi.org/10.7146/math.scand.a-134104Abstract
Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $$ \mathrm{reg} (I(G)^{(s+1)})\leq \max \bigl \{\mathrm{reg} (I(G))$$ $$+2s, \mathrm{reg} \bigl (I(G)^{(s+1)}+I(G)^s\bigr )\bigr \}. $$ As a consequence, we conclude that $\mathrm{reg} (I(G)^{(2)})\leq \mathrm{reg} (I(G))+2$, and $\mathrm{reg} (I(G)^{(3)})\leq \mathrm{reg} (I(G))+4$. Moreover, it is shown that if for some integer $k\geq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $\mathrm{reg} (I(G)^{(s)})\leq 2s+\mathrm{reg} (I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that $\mathrm{reg} (I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline {G}$ is chordal.
References
Alilooee, A., and Banerjee, A., Powers of edge ideals of regularity three bipartite graphs, J. Commut. Algebra 9 (2017), no. 4, 441–454. https://doi.org/10.1216/JCA-2017-9-4-441
Alilooee, A., Beyarslan, S., and Selvaraja, S, Regularity of powers of edge ideals of unicyclic graphs, Rocky Mountain J. Math. 49 (2019), no. 3, 699–728. https://doi.org/10.1216/RMJ-2019-49-3-699
Banerjee, A., The regularity of powers of edge ideals, J. Algebraic Combin. 41 (2015), no. 2, 303–321. https://doi.org/10.1007/s10801-014-0537-2
Banerjee, A., Beyarslan, S., and Hà, H. T., Regularity of edge ideals and their powers, Advances in algebra, 17–52, Springer Proc. Math. Stat., 277, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-11521-0_2
Banerjee, A., Beyarslan, S., and Hà, H. T., Regularity of powers of edge ideals: from local properties to global bounds Algebr. Comb. 3 (2020), no. 4, 839–854. https://doi.org/10.5802/alco.119
Banerjee, A., and Nevo, E., Regularity of edge ideals via suspension, preprint.
Beyarslan, S., Hà, H. T., and Trung, T. N., Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095. https://doi.org/10.1007/s10801-015-0617-y
Cid-Ruiz, Y., Jafari, S., Nemati, N., and Picone, B., Regularity of bicyclic graphs and their powers, J. Algebra Appl. 19 (2020), no. 3, 2050057, 38pp. https://doi.org/10.1142/S0219498820500577
Cutkosky, D., Herzog, J., and Trung, N. V., Asymptotic behaviour of Castelnuovo-Mumford regularity, Compositito Math. 118 (1999), no. 3, 243–261. https://doi.org/10.1023/A:1001559912258
Dao, H., Huneke, C., and Schweig, J. Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), no. 1, 37–55. https://doi.org/10.1007/s10801-012-0391-z
Erey, N., Powers of edge ideals with linear resolutions, Comm. Algebra 46 (2018), no. 9, 4007–4020. https://doi.org/10.1080/00927872.2018.1430810
Erey, N., Powers of ideals associated to $(C_4,2K_2)$-free graphs, J. Pure Appl. Algebra 223 (2019), no. 7, 3071–3080. https://doi.org/10.1016/j.jpaa.2018.10.009
Fröberg, R., On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), 57–70, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.
Gu, Y., Regularity of powers of edge ideals of some graphs, Acta Math. Vietnam. 42 (2017), no. 3, 445–454. https://doi.org/10.1007/s40306-017-0204-5
Y. Gu, Y., Hà, H. T., O'Rourke, J. L., and Skelton, J. W., Symbolic powers of edge ideals of graphs, Comm. Algebra 48 (2020), no. 9, 3743–3760. https://doi.org/10.1080/00927872.2020.1745221
Hà, H. T., Regularity of squarefree monomial ideals, Connections between algebra, combinatorics, and geometry, 251–276, Springer Proc. Math. Stat., 76, Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-0626-0_7
Herzog, J. and Hibi, T., Monomial Ideals, Graduate Texts in Mathematics, 260. Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6
Herzog, J., Hibi, T., and Zheng, X., Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), no. 1, 23–32. https://doi.org/10.7146/math.scand.a-14446
Hoa, L. T., and Tam, N. D., On some invariants of a mixed product of ideals, Arch. Math. (Basel) 94 (2010), no. 4, 327–337. https://doi.org/10.1007/s00013-010-0112-6
Jayanthan, A. V. and Kumar, R., Regularity of symbolic powers of edge ideals, J. Pure Appl. Algebra 224 (2020), no. 7, 106306, 12pp. https://doi.org/10.1016/j.jpaa.2020.106306
Jayanthan, A. V., Narayanan, N., and S. Selvaraja, Regularity of powers of bipartite graphs, J. Algebraic Combin. 47 (2018), no. 1, 17–38. https://doi.org/10.1007/s10801-017-0767-1
Jayanthan, A. V., and Selvaraja, S., Linear polynomial for the regularity of powers of edge ideals of very well-covered graphs, J. Commut. Algebra 13 (2021), no. 1, 89–101. https://doi.org/10.1216/jca.2021.13.89
Katzman, M., Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory, Ser. A 113 (2006), no. 3, 435–454. https://doi.org/10.1016/j.jcta.2005.04.005
Kodiyalam, V., Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. https://doi.org/10.1090/S0002-9939-99-05020-0
Moghimian, M., Seyed Fakhari, S. A., and Yassemi, S., Regularity of powers of edge ideal of whiskered cycles, Comm. Algebra 45 (2017), no. 3, 1246–1259. https://doi.org/10.1080/00927872.2016.1175605
Peeva, I., Graded syzygies, Algebra and Applications, vol. 14, Springer-Verlag London Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-177-6
Rinaldo, G., Terai, N., and Yoshida, K., Cohen-Macaulayness for symbolic power ideals of edge ideals, J. Algebra 347 (2011), 1–22. https://doi.org/10.1016/j.jalgebra.2011.09.007
Seyed Fakhari, S. A., Symbolic powers of cover ideal of very well-covered and bipartite graphs, Proc. Amer. Math. Soc. 146 (2018), no. 1, 97–110. https://doi.org/10.1090/proc/13721
Seyed Fakhari, S. A., Regularity of symbolic powers of edge ideals of unicyclic graphs, J. Algebra 541 (2020), 345–358. https://doi.org/10.1016/j.jalgebra.2019.08.039
Seyed Fakhari, S. A., Regularity of symbolic powers of edge ideals of Cameron-Walker graphs, Comm. Algebra 48 (2020), no. 12, 5215–5223. https://doi.org/10.1080/00927872.2020.1783673
Seyed Fakhari, S. A., Regularity of symbolic powers of edge ideals of chordal graphs, Kyoto J. Math., to appear.
Seyed Fakhari, S. A., and Yassemi, S., Improved bounds for the regularity of edge ideals of graphs, Collect. Math. 69 (2018), no. 2, 249–262. https://doi.org/10.1007/s13348-017-0204-8
Simis, A., Vasconcelos, W., and Villarreal, R. H., On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416. https://doi.org/10.1006/jabr.1994.1192
Woodroofe, R., Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra 6 (2014), no. 2, 287–304. https://doi.org/10.1216/JCA-2014-6-2-287
