Finding Fullerene Patches in Polynomial Time

TL;DR

This paper presents a polynomial-time algorithm for determining if a boundary code corresponds to a fullerene patch with up to five pentagonal faces, advancing understanding of fullerene enumeration.

Contribution

The authors develop a polynomial-time algorithm for recognizing fullerene patches with limited pentagonal faces and propose extending it to count patches with a given boundary code.

Findings

Algorithm works in polynomial time for up to five 5-faces

Conjecture extends the algorithm's correctness to any number of 5-faces

Sketch of extension to counting patches with a given boundary code

Abstract

We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of some fullerene patch can be answered in polynomial time when such patches have at most five 5-faces. We conjecture that our algorithm gives the correct answer for any number of 5-faces, and sketch how to extend the algorithm to the problem of counting the number of different patches with a given boundary code.