Cohen–Macaulay binomial edge ideals of cactus graphs

Abstract

We classify the Cohen-Macaulay binomial edge ideals of cactus and bicyclic graphs.

Cohen–Macaulay binomial edge ideals and accessible graphs

“…Proof Recall that B(G) is a tree by [27,Proposition 1.3], and every cut vertex of G belongs to exactly two blocks of G because J G is unmixed. Let B 1 be a block corresponding to a leaf of B(G).…”

“…Therefore, it is very interesting to find alternative descriptions of Cohen-Macaulayness. In this direction, several authors found constructions [18,24] and described classes of graphs whose binomial edge ideal is Cohen-Macaulay [5,8,26,27]. In this paper, we present the first attempt to find a general combinatorial characterization of Cohen-Macaulay binomial edge ideals, which is only based on the structure of the cut sets of a graph, providing a simpler way to check such homological property.…”

Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

“…Let G be an indecomposable Cohen-Macaulay cactus graph whose blocks are B 1 , • • • , B l . Then it follows from [14,Lemma 2.3] that either G ∈ {K 2 , C 3 } or G satisfies the following conditions:…”

$(S_2)$-condition and Cohen-Macaulay binomial edge ideals

“…There are several attempts at this problem available for some families of graphs. Some papers in this direction are [8], [19], [20], [15], [2], [3], [21], [14], [1], [9], and [4]. In the latter, the authors introduce two combinatorial properties strictly related to the Cohen-Macaulayness of binomial edge ideals: accessibility and strongly unmixedness.…”