Extending patches to fullerenes

Abstract

In this paper we consider fullerene patches that can be extended to pseudconvex patches. We show that all fullerene disks with three or fewer pentagons can be extended to pseudoconvex patches, and that all pseudoconvex patches can be extended to fullerenes.

The saturation number of carbon nanocones and nanotubes

“…Next it will be useful to introduce some definitions utilized in [4,5,[14][15][16]. The boundary code of a patch is described by a sequence of 2's and 3's corresponding to the degree of the vertices on the boundary of the patch in cyclic order.…”

“…The upper bound on the saturation number of nanocones relies on splitting the nanocone in subgraphs, where the number of subgraphs depends on the number of break edges. The following Lemma was proven in [16].…”

“…Lemma 2.4. [16] In a nanocone, the number of pentagons p and the number of break edges s are related by s + p = 6.…”

Coronoids, patches and generalised altans

“…Let G be the subdivided cube in Figure 23. Let C 1 = (1, 2, 3, 4, 7, 6, 5), 1,5,9,10,14,8) and C 3 = (10, 11,12,13,16,15,14). Then (G; C 1 , C 2 , C 3 ) is an admissible structure.…”

“…In a previous paper we shifted attention from benzenoids to the more general subcubic planar graphs that we called patches, which generalise the fullerene patches of Graver et al [11,12,13,14,15,16]. In this paper, we similarly generalise coronoids to perforated patches, i.e., to patches with several disjoint holes.…”