Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11

Let φ P ( C 6 ) (respectively, φ T ( C 6 ) ) be the minimum integer k 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 k . In 1999, S. Jendrol’ and T. Madaras proved that 10 ≤ φ T ( C 6 ) ≤ 11 . It is also known, due to B. Mohar, R. Škrekovski and H.-J. Voss (2003), that φ P ( C 6 ) ≤ 107 . ve that φ P ( C 6 ) = φ T ( C 6 ) = 11 .