Quasi-forest simplicial complexes and almost Cohen-Macaulay

TL;DR

This paper investigates quasi-forest simplicial complexes, introduces new concepts of simplicial cycles and points, and characterizes when these complexes are almost Cohen-Macaulay, including specific results for cycle graphs.

Contribution

It defines simplicial $k$-cycles and points, characterizes quasi-forest complexes via these concepts, and determines conditions for almost Cohen-Macaulay property.

Findings

Quasi-forest complexes lack certain simplicial cycles and points for $k\, extgreater= 3$.

Characterization of almost Cohen-Macaulay quasi-forest complexes.

Cycle graph $C_n$ is almost Cohen-Macaulay only for specific $n$.

Abstract

In this paper we study the quasi-forest simplicial complexes and we define the concept of simplicial $k$-cycle (denoted by $\mathcal{S}_k$) and simplicial $k$-point (denoted by $\mathcal{P}_k$). We show that a simplicial complex $\Delta$ is quasi-forest if and only if it does not have any $\mathcal{P}_k$ and any $\mathcal{S}_k$ for $k\geq 3$. Furthermore we characterize almost Cohen-Macaulay quasi-forest simplicial complexes. In the end we show that the cycle graph $G=C_n$ is almost Cohen-Macaulay if and only if $n=3,4,5,6,7,8,9,11$.