In [10], Hibi and Mahmood introduced a sequence of simplicial complexes associated with a given simplicial complex \(\Omega \) on the vertex set \([n] = \{1, 2, \ldots , n\}\) , where each term in the sequence is obtained by taking the Stanley–Reisner complex of the facet ideal of the previous term. They proved that this sequence eventually returns to the original complex \(\Omega \) after a finite number of steps and continues cyclically thereafter. The smallest such number \(q\) for which the \(\mathcal{N}\mathcal{F}\) -complex is isomorphic to simplicial complex \(\Omega \) , that is, \(\gamma ^{(q)}_{\mathcal{N}\mathcal{F}}(\Omega ) \cong \Omega \) , is called the \(\mathcal{N}\mathcal{F}\) -number of \(\Omega \) . In this paper, we show that the \(\mathcal{N}\mathcal{F}\) -number of \(m\) disjoint copies of the complete graph \(K_n\) is \(mn + 2\) .
