Hyperbolic Analogues of Fullerenes on Orienatable Surfaces

Mathematical models of fullerenes are cubic planar maps with pentagonal and hexagonal faces. As a consequence of Eulers’s formula the number of pentagons in such a map is 12. Conversly, for any integer α ≥ 0, there exists a fullerene map with precisely α hexagons unless α = 1. In this paper, we consider hyperbolic analogues of fullerenes, which are defined as cubic maps of face-type (6, k) on orientable surface of higher genus greater than 1, where by a map of face-type (6, k) we mean a map with only two face lengths: 6 and k for some k ≥ 7. It follows from Euler’s formula that if k is an integer such that for any g ≥ 2 there exists a cubic map of face-type (6, k) and genus g, then k ∈ {7, 8, 9, 10, 12, 18}. In such a map, the number of k-gons is determined in terms of genus, with no condition on the number of hexagons. We show that for any k ∈ {7, 8, 9, 12, 18} and any g ≥ 2 there exists a cubic map of facetype (6, k) with any prescribed number of hexagons. Furthermore, for k = 7 and 8 we prove the existence of polyhedral cubic maps of face-type (6, k) on surfaces of any prescribed genus g ≥ 3 and with any number of hexagons α, with possible exceptions when k = 8 and either g = 2 and α = 4 or g = 3 and α = 1, 2. Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics, Radlinského 11, 813 68, Bratislava, Slovakia, [email protected]. Faculty of Mathematics and Physics, Univerity of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 21, 1111 Ljubljana, Slovenia, [email protected]. Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics, Radlinského 11, 813 68, Bratislava, Slovakia, [email protected]. Faculty of Mathematics and Physics, Univerity of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 21, 1111 Ljubljana, Slovenia, [email protected].