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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.

In this paper, we show that fullerene patches with nice boundaries containing between 1 and 5 pentagons fall into several equivalence classes; furthermore, any two fullerene patches in the same class can be transformed into the same minimal configuration using combinatorial alterations.

For n≥3, a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If n≠3, then such a graph has 2n+1 vertices and 2n+2 edges. In the special case when n=3, a smallest graph has 16 vertices but 44 edges.

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