Let (respectively, ) be the minimum integer with the property that every 3-polytope (respectively, every plane triangulation) with minimum degree 5 has a 6-cycle with all vertices of degree at most . In 1999, S. Jendrol’ and T. Madaras proved that . It is also known, due to B. Mohar, R. Škrekovski and H.-J. Voss (2003), that .