Analyzing all 3-regular graphs that have only faces with 5 edges or more (simplified), I empirically found (using a computer program) that many hypothetically possible graphs, that by Euler’s identity may exist (), do not actually exist. Using a VF2 algorithm to filter out isomorphic maps, I also noticed that not so many graphs as I expected exist. And that one general category of graphs, that always represents a simplified 3-regular graph, is that of Fullerenes (with 12 faces F5 and an arbitrary number of F6). Here is a list of what I found, so far, for each class of graphs, from 12 faces to 20 faces (surrounding area included).

**The question is**: Since the computation of maps with 17, 18, 19, 20 faces (simplified and not containing isomorphic graphs) it is taking very long time (days of CPU time on a PC), is this sequence already known?

I did post this question on: math.stackexchange.com

- 12 faces: 1 (only 1 graph exists)
- On 3 dimentional space (sphere) it is a dodecahedron
- It is a fullerene: 20-fullerene Dodecahedral graph

- 13 faces: 0 (no simplified graphs exist with 13 faces)
- The hypothetical (by Euler’s identity) map of 12 F5 and 1 F6
**does not exist**

- The hypothetical (by Euler’s identity) map of 12 F5 and 1 F6
- 14 faces: 1 (12 F5 + 2 F6)
- The hypothetical (by Euler’s identity) map of 13 F5 and 1 F7
**does not exist** - It is a fullerene: GP (12,2) Generalized Petersen graph

- The hypothetical (by Euler’s identity) map of 13 F5 and 1 F7
- 15 faces: 1 (12 F5 + 3 F6)
- The hypothetical (by Euler’s identity) map of 14 F5 and 1 F8
**does not exist** - It is a fullerene: 26-Fullerene

- The hypothetical (by Euler’s identity) map of 14 F5 and 1 F8
- 16 faces: 3 (Two graphs are 12 F5 + 4 F6. The other has 14 F5 + 2 F7)
- The hypothetical (by Euler’s identity) map of 14 F5 and 2 F7
**does exists** - The other two are Fullerenes

- The hypothetical (by Euler’s identity) map of 14 F5 and 2 F7
- 17 faces: ???
- 18 faces: ???
- 19 faces: ???
- 20 faces: ???

Many fullerenes are here: http://hog.grinvin.org/Fullerenes.

Last things implemented in the Java program (sourceforge):

- Save and restore all lists (maps and todoList) to disk
- Done: 19/Gen/2013

- Verify the various filters when generating maps
- Done: 19/Gen/2013

- Browse the todoList
- Done: 20/Gen/2013

Here are the maps … four colored … up to 16 faces (ocean included):

12 | 14 | 15 |

16 | 16 | 16 |

And the same graphs elaborated using yWorks:

12 | 14 | 15 |

16 | 16 | 16 |