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 $k=2,3,\dots$\,, we introduce {\em $k^{th}$ $\mathcal{N}\mathcal{F}$-complex} ${\delta }_{\mathcal{N}\mathcal{F}}^{(k)}(\Delta )$ which is inductively defined by ${\delta }_{\mathcal{N}\mathcal{F}}^{(k)}(\Delta )={\delta }_{\mathcal{N}\mathcal{F}}({\delta }_{\mathcal{N}\mathcal{F}}^{(k-1)}(\Delta ))$ with setting ${\delta }_{\mathcal{N}\mathcal{F}}^{(1)}(\Delta )={\delta }_{\mathcal{N}\mathcal{F}}(\Delta )$. One can set ${\delta }_{\mathcal{N}\mathcal{F}}^{(0)}(\Delta )=\Delta$. The $\mathcal{N}\mathcal{F}$-number of $\Delta$ is the smallest integer $k>0$ for which ${\delta }_{\mathcal{N}\mathcal{F}}^{(k)}(\Delta )\simeq \Delta$. 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 $n\ge 2$ and $m\ge 2$ with $(n,m)\ne (2,2)$, 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 $n+m+2$.

7 Pages