Bounds on the number of cycles of length three in a planar graph

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF1,F2, …,Fr. Letf1,f2, …,fr denote the lengths of the cycles bounding the facesF1,F2, …,Fr respectively. LetC3(G) be the number of cycles of length three inG. We give bounds onC3(G) in terms ofp,f1,f2, …,fr. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC3(G) between the maximum and minimum value are actually achieved.