The $\mathcal{N}\mathcal{F}$-Number of a Simplicial Complex

Abstract

Let $\Delta$ be a simplicial complex on $[n]$. The $\mathcal{N}\mathcal{F}$-complex of $\Delta$ is the simplicial complex $\delta_{\mathcal{N}\mathcal{F}}(\Delta)$ on $[n]$ for which the facet ideal of $\Delta$ is equal to the Stanley--Reisner ideal of $\delta_{\mathcal{N}\mathcal{F}}(\Delta)$. Furthermore, for each $k = 2,3,\ldots$\,, we introduce {\em $k^{th}$ $\mathcal{N}\mathcal{F}$-complex} $\delta^{(k)}_{\mathcal{N}\mathcal{F}}(\Delta)$ which is inductively defined by $\delta^{(k)}_{\mathcal{N}\mathcal{F}}(\Delta) = \delta_{\mathcal{N}\mathcal{F}}(\delta^{(k-1)}_{\mathcal{N}\mathcal{F}}(\Delta))$ with setting $\delta^{(1)}_{\mathcal{N}\mathcal{F}}(\Delta) = \delta_{\mathcal{N}\mathcal{F}}(\Delta)$. One can set $\delta^{(0)}_{\mathcal{N}\mathcal{F}}(\Delta) = \Delta$. The $\mathcal{N}\mathcal{F}$-number of $\Delta$ is the smallest integer $k > 0$ for which $\delta^{(k)}_{\mathcal{N}\mathcal{F}}(\Delta) \simeq \Delta$. In the present paper we are especially interested in the $\mathcal{N}\mathcal{F}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathcal{N}\mathcal{F}$-number of the finite graph $K_n\coprod K_m$ on $[n + m]$, which is the disjoint union of the complete graphs $K_n$ on $[n]$ and $K_m$ on $[m]$, where $n \geq 2$ and $m \geq 2$ with $(n,m) \neq (2,2)$, is equal to $n + m + 2$. Its corollary says that the $\mathcal{N}\mathcal{F}$-number of the complete bipartite graph $K_{n,m}$ on $[n+m]$ is also equal to $n + m + 2$.