Hyperbolic analogues of fullerenes on orientable surfaces

• Published in: Discrete mathematics 2012-02, Vol.312 (4), p.729-736
• Main Authors: Sikirić, M. Dutour, Knor, M., Potočnik, P., Širáň, J., Škrekovski, R.
• Format: Article
• Language: English
• Subjects: Cubic map; Fullerene; Fullerenes; Hexagons; Integers; Mathematical models; Orientable map; Polyhedral map; Polyhex
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2011.11.009

Mathematical models of fullerenes are cubic spherical maps of 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 maps of face-type ( 6 , k ) for some k ≥ 7 on an orientable surface of genus at least 2. The number of k -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 k that are ‘universal’ in the sense that there exist cubic maps of face-type ( 6 , k ) for all genera g ≥ 2 . By Euler’s formula, if k is universal, then k ∈ { 7 , 8 , 9 , 10 , 12 , 18 } . We show that for any k ∈ { 7 , 8 , 9 , 12 , 18 } and any g ≥ 2 there exists a cubic map of face-type ( 6 , k ) with any prescribed number of hexagons. For k = 7 and 8 we also prove the existence of polyhedral cubic maps of face-type ( 6 , k ) on surfaces of any prescribed genus g ≥ 2 and with any number of hexagons α , except for the cases k = 8 , g = 2 and α ≤ 2 , where we show that no such maps exist.