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

Abstract

We describe the simplicial complex ∆ such that the initial ideal of J G is the Stanley-Reisner ideal of ∆. By ∆ we show that if J G is (S 2 ) then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen-Macaulay are all and only the accessible ones. https://mathworld.wolfram.com/HelmGraph.html.

Cohen-Macaulay generalized binomial edge ideals

“…Hence, we have that |C(G)| = in 2 = 12. Modifying the algorithm implemented in [19] and [16] for the unmixedness of binomial edge ideals we computed all the unmixed J G,m with n ≤ 10 vertices. The implementation and results are downloadable from [1].…”

Cohen-Macaulay Binomial edge ideals in terms of blocks with whiskers

“…We are interested to classify those G for which J G is Cohen-Macaulay. Although, several works have been done in this direction (see [1], [3], [4], [5], [8], [9], [12], [17], [15], [21], [22]), but full characterization of Cohen-Macaulay binomial edge ideals is still widely open.…”

“…Moreover, they conjectured [5,Conjecture 1.1] on the equivalency of these three properties. In [17], the authors showed if R/J G satisfies Serre's condition S 2 , then G is accessible. For any ideal I ⊆ R, it is an well known result that R/I is Cohen-Macaulay if and only if R/I satisfies Serre's condition S r for all r ≥ 1.…”

“…A block of a connected graph G is a maximal induced subgraph of G which has no cut vertex. In many papers, we have seen examples of Cohen-Macaulay J G are some blocks with whiskers (see [4], [5], [17], [21], [22]). We give the motivation to study only blocks with some whiskers to characterize all Cohen-Macaulay J G .…”

“…In Section 2, we recall some definitions, concept, notations related to graph theory and commutative algebra. Then we mention some results from [5], [17], [20] which have been used frequently in our work.…”