Cohen–Macaulay binomial edge ideals in terms of blocks with whiskers

Abstract

For a graph [Formula: see text] and the binomial edge ideal [Formula: see text] of [Formula: see text], Bolognini et al. have proved the following: [Formula: see text] is strongly unmixed [Formula: see text][Formula: see text] is Cohen–Macaulay [Formula: see text][Formula: see text] is accessible. Moreover, they have conjectured that the converse of these implications is true. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all [Formula: see text] with Cohen–Macaulay [Formula: see text]. We show that accessible and strongly unmixed properties of [Formula: see text] depend only on the corresponding properties of its blocks with whiskers and vice versa. We give a new family of graphs whose binomial edge ideals are Cohen–Macaulay, and from that family, we classify all [Formula: see text]-regular [Formula: see text]-connected graphs, with the property that, after attaching some special whiskers to it, the binomial edge ideals become Cohen–Macaulay. To prove the Cohen–Macaulay conjecture, it is enough to show that every non-complete accessible graph [Formula: see text] has a cut vertex [Formula: see text] such that [Formula: see text] is accessible. We show that any non-complete accessible graph [Formula: see text] having at most three cut vertices has a cut vertex [Formula: see text] for which [Formula: see text] is accessible.