Improved bounds for the regularity of edge ideals of graphs

• Autores: S. A. Seyed Fakhari, Siamak Yassemi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 2, 2018, págs. 249-262
• Idioma: inglés
• DOI: 10.1007/s13348-017-0204-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let G be a graph with n vertices, let S={\mathbb {K}}[x_1,\dots ,x_n] be the polynomial ring in n variables over a field {\mathbb {K}} and let I(G) denote the edge ideal of G. For every collection {\mathcal {H}} of connected graphs with K_2\in {\mathcal {H}}, we introduce the notions of {{\mathrm{ind-match}}}_{{\mathcal {H}}}(G) and {{\mathrm{min-match}}}_{{\mathcal {H}}}(G). It will be proved that the inequalities {{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then \mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G). Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then \mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality \mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ]) holds.

• Referencias bibliográficas
  • Athanasiadis, C.: Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat. 49, 17–29 (2011)
  • Biermann, J., Van Tuyl, A.: Balanced vertex decomposable simplicial complexes and their h-vectors. Electron. J. Combin. 20(3), 15 (2013)
  • Biyikŏglu, T., Civan, Y.: Vertex decomposable graphs, codismantlability, Cohen–Macaulayness and Castelnuovo–Mumford regularity. Electron....
  • Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
  • Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38,...
  • Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
  • Fröberg, R.: A note on the Stanley–Reisner ring of a join and of a suspension. Manuscr. Math. 60, 89–91 (1988)
  • Hà, H.T.: Regularity of squarefree monomial ideals. In: Copper, S.M., Sather-Wagstaff, S. (eds.) Connections Between Algebra, Combinatorics,...
  • Hà, H.T., Van Tuyl, A.: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebraic Combin. 27, 215–245 (2008)
  • Hà, H.T., Woodroofe, R.: Results on the regularity of square-free monomial ideals. Adv. Appl. Math. 58, 21–36 (2014)
  • Herzog, J.: A generalization of the Taylor complex construction. Commun. Algebra 35, 1747–1756 (2007)
  • Hibi, T., Higashitani, A., Kimura, K., Tsuchiya, A.: Dominating induced matchings of finite graphs and regularity of edge ideals. J. Algebraic...
  • Hoang, D.T., Minh, N.C., Trung, T.N.: Cohen–Macaulay graphs with large girth. J. Algebra Appl. 14(7), 1550112 (2015). (16 pp)
  • Hoang, D.T., Trung, T.N.: A characterization of triangle–free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals. J....
  • Kalai, G., Meshulam, R.: Intersections of Leray complexes and regularity of monomial ideals. J. Combin. Theory Ser. A 113, 1586–1592 (2006)
  • Katzman, M.: Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A 113, 435–454 (2006)
  • Khosh-Ahang, F., Moradi, S.: Regularity and projective dimension of edge ideal of C_5-free vertex decomposable graphs. Proc. Am. Math. Soc....
  • Kummini, M.: Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebraic Combin. 30, 429–445 (2009)
  • Mahmoudi, M., Mousivand, A., Crupi, M., Rinaldo, G., Terai, N., Yassemid, S.: Vertex decomposability and regularity of very well-covered graphs....
  • Peeva, I.: Graded Syzygies, Algebra and Applications, vol. 14. Springer, London (2011)
  • Van Tuyl, A.: Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity. Arch. Math. (Basel) 93, 451–459 (2009)
  • Walker, J.W.: Topology and combinatorics of ordered sets. Ph.D. Thesis, MIT (1981)
  • Woodroofe, R.: Matchings, coverings, and Castelnuovo–Mumford regularity. J. Commut. Algebra 6, 287–304 (2014)