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
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• Resumen

  • Let G be a graph with n vertices, let ?=?[?1,…,??] be the polynomial ring in n variables over a field ? and let I(G) denote the edge ideal of G. For every collection of connected graphs with ?2∈ , we introduce the notions of ind−match(?) and min−match(?) . It will be proved that the inequalities ind−match{?2,?5}(?)≤reg(?/?(?))≤min−match{?2,?5}(?) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then reg(?/?(?))=ind−match{?2,?5}(?) . Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then reg(?/?(?))=ind−match{?2,?5}(?) 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 reg(?[Δ])=dim(?[Δ]) holds.