Light C4C4 and C5C5 in 3-polytopes with minimum degree 5

Let wP(Cl)wP(Cl) (wT(Cl)wT(Cl)) be the minimum integer kk with the property that every 3-polytope (respectively, every plane triangulation) with minimum degree 5 has an ll-cycle with weight, defined as the degree-sum of all vertices, at most kk.In 1998, O.V. Borodin and D.R. Woodall proved wT(C4)=25wT(C4)=25 and wT(C5)=30wT(C5)=30. We prove that wP(C4)=26wP(C4)=26 and wP(C5)=30wP(C5)=30.