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 parameters exists.