Fullerene patches I

Abstract

Large carbon molecules, discovered at the end of the last century, are called fullerenes. The most famous of these, C 60 has the structure of the soccer ball: the seams represent chemical bonds and the points where three seams come together represent the carbon atoms. We use the term fullerene to represent all mathematically possible structures for the chemical fullerenes: trivalent plane graphs with only hexagonal and pentagonal faces. A simple consequence of Euler's formula is that each fullerene has exactly 12 pentagonal faces; the only restriction on the number of hexagonal faces is that it not be 1. The chemical motivation for this paper is to answer the question: When is it possible to alter a fullerene by changing the structure inside a region of the fullerene bounded by a simple closed circuit? Such a region or patch is said to be ambiguous if alterations may be made to its interior without disturbing the structure of the fullerene outside of the region. In this paper, we show that, relative to the minimum distance between pentagonal faces, there are no small ambiguous patches.

Coronoids, patches and generalised altans

“…We will present a mathematical formalisation which is based on the treatment of coronoids and benzenoids in the previous section of this paper. As we will see later, our definition of a fullerene patch is compatible with Graver's definition [12,13]. There is also a notion of a (m, k)-patch which received a lot of attention in the past years [2,3,11,18].…”

“…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.…”

Fullerene patches II

“…In [7] it was shown that the similarity class of a linear patch with at most one pentagonal end is trivial. On the other hand, the similarity class of any linear patch with two pentagonal ends includes some additional nonlinear patches and is nontrivial, as is illustrated in 1.…”

“…We will adopt the notation from [7] and let a patch be denoted as Π = (V, E, F, B) where V and E are the vertex and edge sets, F is the set of faces excluding the external face, and B is the boundary of the external face. One method of constructing a patch is to take a simple closed curve on a fullerene, and consider the subgraph created by deleting all vertices and edges on the "outside" of the curve.…”