Sequentially $S_r$ simplicial complexes and sequentially $S_2$ graphs

TL;DR

This paper introduces the concept of sequentially $S_r$ modules and simplicial complexes, generalizing Cohen-Macaulay and Serre's conditions, with applications to graph theory and characterizations of specific cycles.

Contribution

It defines sequentially $S_r$ properties for modules and complexes, characterizes these properties algebraically and combinatorially, and extends results to graphs, especially bipartite and cycle graphs.

Findings

Sequentially $S_r$ complexes are characterized by their pure skeletons.

Only odd cycles are sequentially $S_2$ among cycles.

Bipartite graphs are vertex decomposable if and only if they are sequentially $S_2$.

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 $r\ge 3$, 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.