Hyperbolic analogues of fullerenes with face-types (6, 9) and (6, 10)

Mathematical models of fullerenes are cubic polyhedral and spherical maps of face-type (5, 6), that is, with pentagonal and hexagonal faces only. Any such map necessarily contains exactly 12 pentagons, and it is known that for any integer α ≥ 0 except α = 1 there exists a fullerene map with precisely α hexagons. In this paper we consider hyperbolic analogues of fullerenes, modelled by cubic polyhedral maps of face-type (6, k), where k ∈ {9, 10}, on orientable surface of genus at least two. The number of k-gons in this case depends on the genus but the number of hexagons is again independent of the surface. For every triple k ∈ {9, 10}, g ≥ 2 and α ≥ 0, we determine if there exists a cubic polyhedral map of face-type (6, k) with exactly α hexagons on an orientable surface of genus g. The only unsolved cases are k = 10, g = 5 and α ≤ 3 when we are not able to say if a hyperbolic fullerene with these