Sequentially $S_{r}$ simplicial complexes and sequentially $S_{2}$ graphs

SEQUENTIALLY Sr SIMPLICIAL COMPLEXES AND SEQUENTIALLY S2 GRAPHS

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre’s condition Sr . In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton...

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the h-triangle, a doubly-indexed generalization of the h-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generaliz...

Sequentially Cohen-Macaulay Graphs of Form θ_{n1, ...nk}

On Sequentially Cohen-macaulay Complexes and Posets

The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for ...

Whiskers and sequentially Cohen-Macaulay graphs

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...