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- W1747203742 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 Serreâs 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 $rge 3$, 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." @default.
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- W1747203742 date "2010-11-18" @default.
- W1747203742 modified "2023-09-27" @default.
- W1747203742 title "Sequentially $S_{r}$ simplicial complexes and sequentially $S_{2}$ graphs" @default.
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- W1747203742 doi "https://doi.org/10.1090/s0002-9939-2010-10646-9" @default.
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