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

Abstract

We introduce sequentially S r modules over a commutative graded ring and sequentially S r simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition S r . In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially S r if and only if its pure i-skeleton is S r for all i. For r = 2, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially S r if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first r steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially S r cycles showing that the only sequentially S 2 cycles are odd cycles and, for r > 3, no cycle is sequentially S r with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially S r graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially S 2 . We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S 2 . Finally, we propose some questions.