Binomial edge ideals and bounds for their regularity
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Binomial edge ideals and bounds for their regularity Arvind kumar1 Received: 3 January 2019 / Accepted: 4 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let G be a simple graph on n vertices and JG denote the corresponding binomial edge ideal in S = K [x1 , . . . , xn , y1 , . . . , yn ]. We prove that the Castelnuovo–Mumford regularity of JG is bounded above by c(G)+1, when G is a quasi-block graph or semiblock graph. We give another proof of Saeedi Madani–Kiani regularity upper bound conjecture for chordal graphs. We obtain the regularity of binomial edge ideals of Jahangir graphs. Later, we establish a sufficient condition for Hibi–Matsuda conjecture to be true. Keywords Binomial edge ideal · Castelnuovo–Mumford regularity · Chordal graph · Quasi-block graph · Semi-block graph · H-polynomial Mathematics Subject Classification 13D02 · 05E40
1 Introduction Let G be a simple graph on [n] and S = K [x1 , . . . , xn , y1 , . . . , yn ], where K is a field. The binomial edge ideal of the graph G, JG = (xi y j − x j yi : {i, j} ∈ E(G), i < j), was introduced by Herzog et al. in [9] and independently by Ohtani in [21]. Since then researchers have been trying to study the algebraic invariants of JG in terms of the combinatorial invariants of G. In [5,9,13,15,19,23,25,26], the authors have established connections between homological invariants such as depth, codimension, Betti numbers and Castelnuovo–Mumford regularity of JG with certain combinatorial invariants associated with the graph G. The study of Castelnuovo–Mumford regularity of binomial edge ideals has attracted a lot of attention in the recent past due to its algebraic and geometric importance. In [19, Theorem 1.1], Matsuda and Murai proved that for any graph G on [n], l(G) ≤ reg(S/JG ) ≤ n − 1, where l(G) is the length of a longest induced path in G. In the same article, they conjectured that reg(S/JG ) = n−1
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Arvind kumar [email protected] Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
123
Journal of Algebraic Combinatorics
if and only if G = Pn . This conjecture was settled in affirmative by Kiani and Saeedi Madani in [15]. For a graph G, let c(G) denote the number of maximal cliques in G. If G is a closed graph, i.e., the generating set of JG is a Gröbner basis with respect to lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn , then Saeedi Madani and Kiani [25] proved that reg(S/JG ) ≤ c(G). In [26], the following conjecture was proposed. Conjecture 1.1 Let G be a graph on [n]. Then, reg(S/JG ) ≤ c(G). In [7], Ene and Zarojanu proved the conjecture for block graphs. In [12], Jayanthan and Kumar proved the conjecture for k-fan graph of the complete graph. In [24], Rouzbahani Malayeri et al. proved the conjecture for the class of chordal graphs. Recently, in [14], Kahle and Krüsemann proved the conjecture for cographs. In the third section, we prove Saeedi Madani–Kiani conjecture for some classes of nonchordal graphs. We prove Conjecture 1.1 for th
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