OFFSET
6,5
COMMENTS
We count here plane graphs, i.e., graphs embedded in the plane, up to embedding-preserving isomorphism, while such sequences as A003094 count planar graphs (counted up to abstract isomorphism). In this we follow the nomenclature of Brendan McKay, cf. link.
LINKS
B. D. McKay, Plane graphs (see also the section on Planar Graphs on this page).
Eric Weisstein's World of Mathematics, Icosahedral Graph.
EXAMPLE
There is only a(6) = 1 planar 5-regular simple connected graph with 2n = 12 vertices, which is the icosahedral graph, cf. MathWorld link. If we label the vertices 1, ..., 9, A, B, C, they are connected as follows: 1 -> {2 3 4 5 6}, 2 -> {1 6 7 8 3}, 3 -> {1 2 8 9 4}, 4 -> {1 3 9 A 5}, 5 -> {1 4 A B 6}, 6 -> {1 5 B 7 2 }, 7 -> {2 6 B C 8}, 8 -> {2 7 C 9 3}, 9 -> {3 8 C A 4}, A -> {4 9 C B 5}, B -> {5 A C 7 6}, C -> {7 B A 9 8}.
For other numbers of vertices, the number of plane 5-regular simple connected graphs is as follows:
14 vertices: 0 graphs,
16 vertices: 1 graph,
18 vertices: 1 graph,
20 vertices: 6 graphs,
22 vertices: 14 graphs,
24 vertices: 98 graphs,
26 vertices: 529 graphs,
28 vertices: 4035 graphs,
30 vertices: 31009 graphs,
32 vertices: 252386 graphs,
34 vertices: 2073769 graphs,
36 vertices: 17277113 graphs. (From the McKay web page.)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
M. F. Hasler, Mar 20 2018
STATUS
approved