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Hyperbolic analogues of fullerenes on orientable surfaces

Dutour Sikirić, M.; Knor, M.; Potocnik, P.; Siran, J. and Škrekovski, R. (2012). Hyperbolic analogues of fullerenes on orientable surfaces. Discrete Mathematics, 312(4) pp. 729–736.

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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.

Item Type: Journal Item
Copyright Holders: 2011 Elsevier B.V.
ISSN: 1872-681X
Keywords: fullerene; polyhex; orientable map; cubic map; polyhedral map
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 34907
Depositing User: Jozef Širáň
Date Deposited: 30 Oct 2012 11:39
Last Modified: 07 Dec 2018 10:09
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