Constructions of fullerenes that maximize the anionic Clar number.

Background

A fullerene $F_n$ is a 3-regular graph such that every face is a pentagon or a hexagon. By Euler's formula, there are exactly 12 pentagons. Let $F(F_n)$ and $E(F_n)$ denote the set of faces and edges in a fullerene $F_n$, respectively. For a fixed integer p, a p-anionic resonance structure $(\mathcal{F}, \mathcal{M})$ of a fullerene $F_n$ is a set of independent faces $\mathcal{F} \subseteq F(F_n)$ (containing exactly p pentagons) and a perfect matching $\mathcal{M} \subseteq E(F_n)$ on the graph obtained from $F_n$ by removing the vertices of the faces in $\mathcal{F}$. The p-anionic Clar number $C_p(F_n)$ of a fullerene $F_n$ is defined to be zero if $F_n$ has no p-anionic resonance structures and, otherwise, to be equal to the maximum value of $|\mathcal{F}|$ over all p-anionic resonance structures $(\mathcal{F}, \mathcal{M})$ of $F_n$. A p-anionic resonance structure that has $C_p(F_n)$ faces in $\mathcal{F}$ is called a p-anionic Clar structure on $F_n$.

Code

The code in this repo generates the adjacency lists of fullerenes defined by the constructions in [INSERT PAPER]. These fullerenes are used to prove tight upper bounds for the $p$-anionic Clar number of fullerenes on $n$ vertices (for even $p > 0$).

NOTE: There are other fullerenes that maximize the $p$-anionic Clar numbers on $n$ vertices, the fullerenes generated here are simply those defined by the constructions in [INSERT PAPER].

Requirements:

A Python library that can call import sys

Files:

There are seven files used to generate fullerenes:

build/02_make_adj.py <- Fullerenes that maximize $C_2(F_n)$ on $n \ge 40$ ($n \neq 44$) vertices.

build/04_make_adj.py <- Fullerenes that maximize $C_4(F_n)$ on $n \ge 36$ ($n \neq 40$) vertices.

build/06_make_adj.py <- Fullerenes that maximize $C_6(F_n)$ on $n \ge 56$ vertices.

build/08_make_adj.py <- Fullerenes that maximize $C_8(F_n)$ on $n \ge 58$ vertices.

build/10_make_adj.py <- Fullerenes that maximize $C_{10}(F_n)$ on $n \ge 54$ ($n \neq 58$) vertices.

build/12_2_mod_6_make_adj.py <- Fullerenes that maximize $C_{12}(F_n)$ on $n \ge 110$ ($n \equiv 2 \mod{6}$) vertices.

build/12_4_mod_6_make_adj.py <- Fullerenes that maximize $C_{12}(F_n)$ on $n = 70$ and $n \ge 88$ ($n \equiv 4 \mod{6}$) vertices.

Note: Leapfrog fullerenes maximize the 12-anionic Clar number $n \equiv 0 \mod{6}$ vertices.

To run:

python {file you want to run} {n}

where n is the maximum size of a fullerene you would like to generate.

Output:

The adjacency lists of fullerenes that maximize the chosen $p$-anionic Clar number with the following format:

{number of vertices in graph (call it n)}
{degree of vertex 0} {neighbor 0} {neighbor 1} {neighbor 2}
{degree of vertex 1} {neighbor 0} {neighbor 1} {neighbor 2}
...
{degree of vertex n-1} {neighbor 0} {neighbor 1} {neighbor 2}

Such that there exists a planar embedding of the vertices where each neighbor is listed in clockwise order.

NOTE: The constructions implemented in the code do not necessarily construct unique fullerenes. To check that two fullerenes on $n$ vertices (with the output adjacency lists) are unique, you need to check that there is no automorphism between the two.

Citation

If you use this code in your research, please cite it via:

@software{Slobodin_Constructions_of_fullerenes_2024,
  author =        {Slobodin, A.},
  month =         oct,
  title =         {{Constructions of fullerenes that maximize the
                   anionic Clar number}},
  year =          {2024},
  url =           {https://github.com/fastbodin/anionic_clar_fullerene_construction},
}