Hyperbolic analogues of fullerenes on orientable surfaces

Mathematical models of fullerenes are cubic spherical maps of type (5,6)(5,6), that is, with pentagonal and hexagonal faces only. Any such map necessarily contains exactly 1212 pentagons, and it is known that for any integer α≥0α≥0 except α=1α=1 there exists a fullerene map with precisely αα hexagons.In this paper we consider hyperbolic analogues of fullerenes, modelled by cubic maps of face-type (6,k)(6,k) for some k≥7k≥7 on an orientable surface of genus at least 2. The number of kk-gons in this case depends on the genus but the number of hexagons is again independent of the surface. We focus on the values of kk that are ‘universal’ in the sense that there exist cubic maps of face-type (6,k)(6,k) for all   genera g≥2g≥2. By Euler’s formula, if kk is universal, then k∈{7,8,9,10,12,18}k∈{7,8,9,10,12,18}.We show that for any k∈{7,8,9,12,18}k∈{7,8,9,12,18} and any g≥2g≥2 there exists a cubic map of face-type (6,k)(6,k) with any prescribed number of hexagons. For k=7k=7 and 88 we also prove the existence of polyhedral   cubic maps of face-type (6,k)(6,k) on surfaces of any prescribed genus g≥2g≥2 and with any number of hexagons αα, except for the cases k=8k=8, g=2g=2 and α≤2α≤2, where we show that no such maps exist.