The height of faces of $3$-polytopes

The height of a face in a $3$-polytope is the maximum degree of the incident vertices of the face, and the height of a $3$-polytope, $h$, is the minimum height of its faces. A face is pyramidal if it is either a $4$-face incident with three $3$-vertices, or a $3$-face incident with two vertices of degree at most $4$. If pyramidal faces are allowed, then $h$ can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated $3$-polytope has $h\le11$. In 1995, this bound was lowered by Avgustinovich and Borodin to $10$. Recently, we improved it to the sharp bound $8$. For plane triangulation without $4$-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that $h\le20$ which bound is sharp. Later, Borodin (1998) proved that $h\le20$ for all triangulated $3$-polytopes. Recently, we obtained the sharp bound $10$ for triangle-free $3$-polytopes. In 1996, Horňák and Jendrol' proved for arbitrarily $3$-polytopes that $h\le23$. In this paper we improve this bound to the sharp bound $20$.