Light and in 3-polytopes with minimum degree 5

Let w P ( C l ) ( w T ( C l ) ) be the minimum integer k with the property that every 3-polytope (respectively, every plane triangulation) with minimum degree 5 has an l -cycle with weight, defined as the degree-sum of all vertices, at most k . 8, O.V. Borodin and D.R. Woodall proved w T ( C 4 ) = 25 and w T ( C 5 ) = 30 . We prove that w P ( C 4 ) = 26 and w P ( C 5 ) = 30 .