## 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.

Original language | English |
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Pages (from-to) | 1993-2005 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 139 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2011 |

Externally published | Yes |

## Keywords

- Sequentially Cohen-Macaualy
- Sequentially Sr simplicial complex
- Serre's condition

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics