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We present a computationally efficient algorithm that is suitable for GPUimplementation. This algorithm enables the identification of all weakpseudo-manifolds that meet specific facet conditions, drawn from a given inputset. We employ this approach to enumerate toric colorable seed PL-spheres.Consequently, we achieve a comprehensive characterization of PL-spheres ofdimension n-1 with n+4 vertices that possess a maximal Buchstaber number. Aprimary focus of this research is the fundamental categorization ofnon-singular complete toric varieties of Picard number 4. This classificationserves as a valuable tool for addressing questions related to toric manifoldswith a Picard number of 4. Notably, we have determined which of these manifoldssatisfy equality within an inequality regarding the number of minimalcomponents in their rational curve space. This addresses a question posed byChen-Fu-Hwang in 2014 for this specific case.

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