A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

Geometric vertex decomposition and liaison

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras.

Symbolic powers of vertex cover ideals

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

A note on the van der Waerden complex

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable.

On Generalization of Cycles and Chordality to Clutters from an Algebraic Viewpoint

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if [Formula: see text] is a clutter such that 〈[Formula: see text]〉 is a vertex decomposable simplicial complex or I([Formula: see text]) is squarefree stable, then [Formula: see text] is chordal.