Binomial edge ideals of bipartite graphs

Abstract

Abstract. We classify the bipartite graphs G whose binomial edge ideal JG is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. We study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with JG unmixed and not Cohen-Macaulay.

Cohen–Macaulay binomial edge ideals of cactus graphs

“…Applying Proposition 1.4, we have that the cutsets in C(G v ) are exactly the cutsets in C(G) not containing v. Moreover the connected components induced by any T in G and in G v have the same set of vertices as stated in the proof of Proposition 1.4 (see (2)). Using the notation introduced in the mentioned proof H i = G i for each connected component, that is…”

“…Since by hypothesis G is indecomposable, two complete graphs cannot be adjacent in B(G). We end observing that if a K 2 is between two cycles C 4 it is not unmixed (see also Remark 4.7 of [2]). (3) ⇒ (1).…”

“…In [4] and [9] the authors considered the Cohen-Macaulay property of these graphs. Recently nice results on Cohen-Macaulay bipartite graphs and blocks have been obtained (see [7], [2] and [1]).…”

Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

“…Observe that C n − n is P m = P n−1 . Set 3]] and R = S/J. We have already proved that J is a prime complete intersection of height m − 1.…”

“…Indeed, the point is that a set of m-minors of a generic matrix m×n does not generate a radical ideal in general (as it does for m = 2). For example, in the Grassmannian G (3,6) Next we look into necessary conditions for I k d (X gen G ) and I k d (X sym G ) to be prime. Lemma 7.10.…”