Alternating plane graphs
An alternating plane graph is a simple, connected plane graph, with minimum degree equal to 3 and where each face has at least 3 sides, in which no pair of adjacent vertices have the same degree, and no pair of adjacent faces have the same number of sides.
For weak alternating plane graphs, we loosen the definition to also allow graphs with minimum degree equal to 2.
The following lists are available:
The graph lists are currently only available in 'planar_code' format. The larger files are compressed with gzip.
Alternating plane graphs
These graphs were constructed by the exhaustive algorithm described in [1] and the program is available at [2]. The numbers were independently verified.
| Vertices | Graphs |
|---|---|
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | 0 |
| 5 | 0 |
| 6 | 0 |
| 7 | 0 |
| 8 | 0 |
| 9 | 0 |
| 10 | 0 |
| 11 | 0 |
| 12 | 0 |
| 13 | 0 |
| 14 | 0 |
| 15 | 0 |
| 16 | 0 |
| 17 | 2 |
| 18 | 0 |
| 19 | 5 |
Weak alternating plane graphs
Below we list the number of weak alternating plane graphs with degrees 2 and k for 3 ≤ k ≥ 8. These graphs were constructed using the technique described in [1] and the programs are available at [2].
| n\k | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| 9 | - | 1 | - | - | - | - |
| 10 | - | - | - | - | - | - |
| 11 | - | - | - | - | - | - |
| 12 | - | 1 | - | - | - | - |
| 13 | - | - | - | - | - | - |
| 14 | - | - | - | - | - | - |
| 15 | - | 2 | - | - | - | - |
| 16 | - | - | - | 1 | - | - |
| 17 | - | - | - | - | - | - |
| 18 | - | 4 | - | - | - | - |
| 19 | - | - | - | - | - | - |
| 20 | 1 | - | - | 0 | - | - |
| 21 | - | 7 | - | - | - | - |
| 22 | - | - | - | - | - | - |
| 23 | - | - | - | - | - | - |
| 24 | - | 19 | - | 1 | - | - |
| 25 | 6 | - | - | - | - | - |
| 26 | - | - | - | - | - | - |
| 27 | - | 43 | - | - | - | - |
| 28 | - | - | 7 | 1 | - | - |
| 29 | - | - | - | - | - | - |
| 30 | 43 | 125 | - | - | - | 1 |
| 31 | - | - | - | - | - | - |
| 32 | - | - | - | 11 | - | - |
| 33 | - | 368 | - | - | - | - |
| 34 | - | - | - | - | - | - |
| 35 | 316 | - | 139 | - | - | 0 |
| 36 | - | 1 264 | - | 10 | 1 | - |
| 37 | - | - | - | - | - | - |
| 38 | - | - | - | - | - | - |
| 39 | - | 4 744 | - | - | - | - |
| 40 | 2 420 | - | - | 83 | - | 1 |
| 41 | - | - | - | - | - | - |
| 42 | - | 18 723 | 4 731 | - | - | - |
| 43 | - | - | - | - | - | - |
| 44 | - | - | - | - | - | - |
| 45 | 19 648 | 78 657 | - | - | - | 1 |
| 46 | - | - | - | - | - | - |
| 47 | - | - | - | - | - | - |
| 48 | - | 338 945 | - | - | - | - |
| 49 | - | - | - | - | - | - |
| 50 | 165 724 | - | - | - | - | - |
| 51 | - | 1 518 480 | - | - | - | - |
| 52 | - | - | - | - | - | - |
| 53 | - | - | - | - | - | - |
| 54 | - | - | - | - | - | - |
| 55 | 1 437 049 | - | - | - | - | - |
References
[1] I. Althöfer, J.K. Haugland, K. Scherer, F. Schneider, N. Van Cleemput, Ars Mathematica Contemporanea, 8(2), 337-363, 2015.