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

Mathematical models of fullerenes are cubic polyhedral and spherical maps of face-type (5,6)(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≥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 polyhedral maps of face-type (6,k)(6,k), where k∈{9,10}k∈{9,10}, on orientable surface of genus at least two. The number of kk-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}k∈{9,10}, g≥2g≥2 and α≥0α≥0, we determine if there exists a cubic polyhedral map of face-type (6,k)(6,k) with exactly αα hexagons on an orientable surface of genus gg. The only unsolved cases are k=10k=10, g=5g=5 and α≤3α≤3 when we are not able to say if a hyperbolic fullerene with these parameters exists.