The characterization of $\mathbf{(n-1)}$-spheres with $\mathbf{n+4}$ vertices having maximal Buchstaber number

We present a computationally efficient algorithm that is suitable for GPU implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seed PL-spheres. Consequently, we achieve a comprehensive characterization of PL-spheres of dimension n-1 with n+4 vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4. This classification serves as a valuable tool for addressing questions related to toric manifolds with a Picard number of 4. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen-Fu-Hwang in 2014 for this specific case.