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

Let $\mathrm{\Delta }$ be a simplicial complex on $[n]$. The $\mathcal{N}\mathcal{F}$-complex of $\mathrm{\Delta }$ is the simplicial complex ${\delta }_{\mathcal{N}\mathcal{F}}(\mathrm{\Delta })$ on $[n]$ for which the facet ideal of $\mathrm{\Delta }$ is equal to the Stanley–Reisner ideal of ${\delta }_{\mathcal{N}\mathcal{F}}(\mathrm{\Delta })$. Furthermore, for each $k = 2,3,\ldots\,, we introduce {\emk^{th}$ $\mathcal{N}\mathcal{F}$-complex} ${\delta }_{\mathcal{N}\mathcal{F}}^{(k)}(\mathrm{\Delta })$ which is inductively defined by $\delta^{(k)}{$\mathcal{N}$$\mathcal{F}$}(\Delta) = \delta{$\mathcal{N}$$\mathcal{F}$}(\delta^{(k-1)}{$\mathcal{N}$$\mathcal{F}$}(\Delta))$withsetting$\delta^{(1)}{$\mathcal{N}$$\mathcal{F}$}(\Delta) = \delta_{$\mathcal{N}$$\mathcal{F}$}(\Delta)$\mathrm{.}Onecanset$\delta^{(0)}_{$\mathcal{N}$$\mathcal{F}$}(\Delta) = \Delta$\mathrm{.}The$$\mathcal{N}$$\mathcal{F}$$-numberof$\Delta$isthesmallestinteger$k > 0$ for which ${\delta }_{\mathcal{N}\mathcal{F}}^{(k)}(\mathrm{\Delta })\simeq \mathrm{\Delta }$. 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 $[n + m]$,whichisthedisjointunionofthecompletegraphs$K_n$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 $n+m+2$.