# $k$-shellable simplicial complexes and graphs

## TL;DR

This paper explores the properties of $k$-shellable simplicial complexes and graphs, demonstrating their relation to shellability, face rings, and the Stanley conjecture, and extending existing results in the field.

## Contribution

It proves that $k$-shellable complexes are expansions of shellable complexes and shows their face rings satisfy the Stanley conjecture, extending known results to a broader class.

## Key findings

- $k$-shellable complexes are expansions of shellable complexes
- Face rings of pure $k$-shellable complexes satisfy Stanley conjecture
- Characterizations of $k$-shellable graphs extend previous results

## Abstract

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture. Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrill\'{o}n-Cruz, Cruz-Estrada and Van Tuyl-Villareal.