The classification of complete non-singular toric varieties with small Picard numbers is a fundamental problem in toric geometry. Kleinschmidt completed the classification for Picard number 2 in 1988, followed by Batyrev’s extension to Picard number 3 in 1991. Continuing this line of work, we provide a complete classification for Picard number 4. A key ingredient in our approach is the notion of toric colorability, which involves assigning linearly independent vectors in $\mathbb{Z}_2^n$ to the vertices of a simplicial sphere of dimension $n-1$. This combinatorial condition plays a central role in understanding the possible underlying spaces of fans, and thus in the classification of toric manifolds.
