### Abstract

Given the Kekulé structure of a fullerene that gives its Clar structure, the Kekulé edges that do not lie on any benzene ring of the Clar structure lie on open chains that pair up the pentagonal faces and perhaps some closed chains. In this paper, we introduce Fries chains and show that given the Kekulé structure of a fullerene that gives its Fries structure, the Kekulé edges that do not lie on two benzene rings lie on the union of a set of these Fries chains. The edges that lie on exactly one benzene ring belong to exactly one of these chains while edges that lie on no benzene ring belong to exactly two of these chains. We will see that the Fries chains will include the Clar chains in some cases and will be quite distinct from the Clar chains in other cases. Examples of fullerenes with the property that the set of benzene rings that give its Clar number are not a subset of the benzene rings that give its Fries number have been known for a few years. However, in all of these known examples, almost all of the benzene rings of its Clar structure are among benzene rings of its Fries structure. In this paper, we describe a class of fullerenes with the property that the set of benzene rings of its Clar structure and the set of benzene rings of its Fries structure practically disjoint.

Original language | English (US) |
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Pages (from-to) | 112-125 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 215 |

DOIs | |

State | Published - Dec 31 2016 |

### Keywords

- Clar number
- Conjugated 6-circuits
- Fries number
- Fullerenes
- Kekulé structure
- Perfect matching

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*215*, 112-125. https://doi.org/10.1016/j.dam.2016.07.016