198 research outputs found

    The regularity of binomial edge ideals of graphs

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    In this paper, we study the Castelnuovo-Mumford regularity and the graded Betti numbers of the binomial edge ideals of some classes of graphs. Our special attention is devoted to a conjecture which asserts that the number of maximal cliques of a graph provides an upper bound for the regularity of its binomial edge ideal

    The regularity of binomial edge ideals of graphs

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    We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for graphs which has a cut edge or a simplicial vertex, and hence for chordal graphs. We determine the regularity of the binomial edge ideal of the join of graphs in terms of the regularity of the original graphs, and consequently prove the conjecture of Matsuda and Murai for such a graph, and hence for complete tt-partite graphs. We also generalize some results of Schenzel and Zafar about complete tt-partite graphs. We also prove the conjecture due to the authors for a class of chordal graphs.Comment: 11 pages, 1 figur

    Local cohomology of binomial edge ideals and their generic initial ideals

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    We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen–Macaulayness and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla (J Comb Algebra 2:231–257, 2018) relating the graded components of the local cohomology modules of Cartwright–Sturmfels ideals and their generic initial ideals. We provide an alternative proof for the case of binomial edge idealsPeer ReviewedPostprint (author's final draft
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