### Abstract

It has been shown that the boundary structure of patches with all faces of the same size k, all interior vertices of the same degree m and all boundary vertices of degree at most m determines the number of faces of the patch (Brinkmann et al., Graphs and discovery, 2005; Guo et al., Discrete Appl Math 118(3):209-222, 2002). In case of at least two defective faces, that is faces with degree ḱ ≠ k, it is well known that this is not the case. The most famous example for this is the Endo-Kroto C _{2}-insertion (Endo and Kroto, J Phys Chem 96:6941-6944, 1992). Patches with alimited amount of disorder are especially interesting for the case k = 6, m = 3 and ḱ = 5. This case corresponds to polycyclic hydrocarbons with a limited number of pentagons and to subgraphs of fullerenes. The last open question was the case of exactly one defective face or vertex. In this paper we generalize the results of Brinkmann et al. (2005) and Guo et al. (2002) and in some cases corresponding to Euclidean lattices also deal with patches that have vertices of degree larger than m on the boundary, have sequences of degrees on the boundary that are identical only modulo m and have vertex and face degrees in the interior that are multiples of m, resp. k. Furthermore we prove that in case of at most one defective face with a degree that is not a multiple of k the number of faces of a patch is determined by the boundary. This result implies that fullerenes cannot grow by replacing patches of a restricted size.

Original language | English (US) |
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Pages (from-to) | 263-278 |

Number of pages | 16 |

Journal | Journal of Mathematical Chemistry |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2009 |

### Keywords

- Face degree
- Fullerene
- Patch
- Planar graph
- Vertex degree

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

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## Cite this

*Journal of Mathematical Chemistry*,

*45*(2), 263-278. https://doi.org/10.1007/s10910-008-9403-6