# Question about 3-regular graphs with a restriction (also fullerene and four color theorem)

(Crossposted to mathoverflow.net)

http://mathoverflow.net/questions/120162/question-about-3-regular-graphs-with-a-restriction-also-fullerene-and-four-color

Studying all 3-regular graphs that have only faces with 5 edges or more (simplified), I empirically found (computer program) that many hypothetically possible graphs, that by Euler's identity may exist ($F5 = 12 + F7 + 2F8 + 3F9 + ...$), do not actually exist. Using a VF2 algorithm to filter out isomorphic maps being created, 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?

• 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
• 14 faces: 1 (12 F5 + 2 F6)
• he 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
• 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
• 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
• 17 faces: ???
• 18 faces: ???
• 19 faces: ???
• 20 faces: ???

ADD (28/Jan/2013):

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Have you checked OEIS? BTW +1 cool question. add the chemistry-tag...;-) I would expect 17 and 19 to have no graphs... –  draks ... Jan 25 '13 at 23:32
I did a quick check but for now the sequence 1,0,1,1,3 has 252 match. I suppose still too few elements! –  Mario Stefanutti Jan 25 '13 at 23:39
... OEIS 1,0,1,1,3,0,_,0... –  draks ... Jan 25 '13 at 23:40
With 17 and 19 faces for sure fullerene exist: hog.grinvin.org/Fullerenes –  Mario Stefanutti Jan 25 '13 at 23:41
cool, I just thought about assuming Barnette conjecture and using grinberg's theorem to exclude some cases, but (i) it seems obsolete now and (ii) I gotta go. Good nite... –  draks ... Jan 25 '13 at 23:49