Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

Abstract

Abstract. We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen-Macaulay and unmixed. So that we generalize the results of Ene, Herzog and Hibi on block graphs. Moreover, we study unmixedness and Cohen-Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs'.

On the binomial edge ideals of block graphs

“…Moreover, for m ≥ n, we show that reg J G = reg(in < (J G )) = n, where n is the number of vertices of the graph G. When m < n, then we provide an upper bound for the regularity of in < (J G ) and, therefore, for the regularity of J G as well. Our results generalize the ones obtained in the papers [3,8,12] for classical binomial edge ideals associated with (generalized) block graphs.…”

“…. .,t q , r. Hence A is a (q + 1)-minimal cut set of G. For any cut point set T of G, we have A T if and only if A ∩ T = / 0 (see proof of Theorem 3.2 in [12]).…”

“…This class of graphs was considered in [12]. Obviously, every block graph is a generalized block graph.…”

The Castelnuovo–Mumford regularity of binomial edge ideals

“…Also, one could see this ideal as an ideal generated by a collection of 2-minors of a (2 × n)-matrix whose entries are all indeterminates. Many of the algebraic properties and invariants of such ideals were studied in [1,2,3,4,5,6,7], [11,12,13] and [14]. One of these invariants is the Castelnuovo-Mumford regularity.…”

Binomial edge ideals of bipartite graphs

“…Binomial edge ideals have been extensively studied, see e.g. [1], [5], [6], [13], [14], [15], [18], [19]. Yet a number of interesting questions is still unanswered.…”