On the regularity of binomial edge ideals

Abstract

We study the regularity of binomial edge ideals. For a closed graph G we show that the regularity of the binomial edge ideal JG coincides with the regularity of in lex (JG) and can be expressed in terms of the combinatorial data of G. In addition, we give positive answers to Matsuda‐Murai conjecture for some classes of graphs.

Krull dimension and regularity of binomial edge ideals of block graphs

“…Lower and upper bounds for the regularity are known by Matsuda and Murai [14] and Kiani and Saeedi Madani [12]. In [5], the authors proved the conjecture posed in [20] for closed graphs and block graphs. For these graphs, the regularity of S/J G is bounded below by the length of the longest induced path of G and above by c(G), where c(G) is the number of maximal cliques of G. Furthermore, Kiani and Saeedi Madani characterized all graphs whose binomial edge ideal have regularity 2 and regularity 3, see [19] and [21].…”

The Castelnuovo–Mumford regularity of binomial edge ideals

“…Ene and Zarojanu in [3] proved Conjecture A for block graphs which are included in the class of chordal graphs.…”

Cartwright–Sturmfels ideals associated to graphs and linear spaces

“…Matsuda and Murai [25] proved that the regularity of the binomial edge ideal associated to a graph G is bounded by the number of vertices of the graph and conjectured that equality holds only when the graph is a line. Other interesting results concerning binomial edge ideals can be found, for example, in [17,18,21]. We prove that binomial edge ideals are CS and describe the associated generic initial ideal.…”