Algebraic properties of the path complexes of cycles

# Algebraic properties of the path complexes of cycles

Abstract: Let $G$ be a simple graph and $\Delta_t \left(G\right)$ be a simplicial complex whose facets correspond to the paths of length $t$ $(t\geq2)$ in $G$. It is shown that $\Delta_t \left(C_n\right)$ is matroid, vertex decomposable, shellable and Cohen-Macaulay if and only if $n=t$ or $n=t+1$, where $C_n$ is an $n$-cycle. As a consequence we show that if $n=t$ or $t+1$ then $\Delta_t\left (C_n\right )$ is partitionable and Stanley's conjecture holds for $K[\Delta_t\left(C_n\right)]$.

Keywords: Vertex decomposable, simplicial complex, Matroid, Path.