All 4-regular-plane graphs with given faces

For detailed information on the number of 4-regular plane graphs on $n ~(\le 60)$ vertices with all faces 3-faces or-4-faces, see A111361.

A111361		The number of 4-regular plane graphs with all faces 3-gons or 4-gons.
n		a(n)
2		0
3		0
4		0
5		0
6		1
7		0
8		1
9		1
10		2
11		1
12		5
13		2
14		8
15		5
16		12
17		8
18		25
19		13
20		30
21		23
22		51
23		33
24		76
25		51
26		109
27		78
28		144
29		106
30		218
31		150
32		274
33		212
34		382
35		279
36		499
37		366
38		650
39		493
40		815
41		623
42		1083
43		800
44		1305
45		1020
46		1653
47		1261
48		2045
49		1554
50		2505
51		1946
52		3008
53		2322
54		3713
55		2829
56		4354
57		3418
58		5233
59		4063
60		6234

How to generate them

We use CaGe to generate them. See https://github.com/CaGe-graph/CaGe. If we don't want to use a Java-based frontend, we can choose two files in CaGe named quad_restrict.c and plantri.c to compile. Note that we also compile a binary file for Windows by use cygwin, named by quad_restrict.exe.

cc -o quad_restrict -O '-DPLUGIN="quad_restrict.c"' -DALLTOGETHER plantri.c

then for example, use

./quad_restrict -q -F3F4  26 >quad_24_F3F4.plc

to generate 24-vertex 4-regular plane graphs with with all faces 3-facesor 4-faces. Note that it has 26 faces, and in the program, we use option 26 since it use dual graphs.

Similary,

./quad_restrict -q -F3F4  27 >quad_25_F3F4.plc
./quad_restrict -q -F3F4  29 >quad_27_F3F4.plc

How to read it

1. add an option -g for graph6 format and using showg , for example, ./quad_restrict -q -F3F4 27 -g but it will miss embedding information.
2. use Mathematica (This code should be due to Szabolcs Horvát）

Needs["IGraphM`"]
decode[data_] := 
 Module[{d = data, head, vc, g},(*skip header if it exists*)
  head = ToCharacterCode[">>planar_code le<<"];
  If[Take[d, Min[Length[d], Length[head]]] === head, 
   d = Drop[d, Length[head]];];
  (*split data at zero separators*)d = Most /@ Split[d, #1 =!= 0 &];
  First@Last@
    Reap@While[Length[d] > 0, 
      vc = d[[1, 
        1]];(*vertex count*)(*get as many neighbour lists as the \
vertex count*){g, d} = TakeDrop[d, vc];
      (*drop vertex count from first neighbour list*)
      g = MapAt[Rest, g, {1}];
      (*build association for combinatorial embedding*)
      Sow@AssociationThread[Range[vc], 
        Reverse /@ g (*reverse from clockwise to counterclockwise*)]]]

data = Import[
   "~\\quad_24_F3F4.plc", "Byte"];
G = IGLayoutTutte@IGAdjacencyGraph[#] & /@ decode[data];

Further...

If there exist two triangle faces in a given plane graph that share a common vertex, return False; otherwise, return True.

Findcomm[g_] :=
 Module[{AllF3}, AllF3 = Select[PlanarFaceList[g], Length[#] == 3 &];
   If[Length[
     Cases[AllF3, {___,
       SelectFirst[Tally[Join @@ AllF3], #[[2]] >= 2 &][[1]], ___}]] ==
     2, False, True]]

For example,

G1=Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, {Null, SparseArray[Automatic, {24, 24}, 0, {1, {{0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96}, {{2}, {3}, {4}, {5}, {1}, {5}, {6}, {7}, {1}, {7}, {8}, {9}, {1}, {9}, {10}, {11}, {1}, {2}, {11}, {12}, {2}, {12}, {13}, {14}, {2}, {3}, {8}, {14}, {3}, {7}, {15}, {16}, {3}, {4}, {16}, {17}, {4}, {11}, {17}, {18}, {4}, {5}, {10}, {19}, {5}, {6}, {13}, {19}, {6}, {12}, {20}, {21}, {6}, {7}, {15}, {21}, {8}, {14}, {21}, {22}, {8}, {9}, {17}, {22}, {9}, {10}, {16}, {23}, {10}, {19}, {20}, {23}, {11}, {12}, {18}, {20}, {13}, {18}, {19}, {24}, {13}, {14}, {15}, {24}, {15}, {16}, {23}, {24}, {17}, {18}, {22}, {24}, {20}, {21}, {22}, {23}}}, Pattern}]}, {FormatType -> TraditionalForm, GraphLayout -> {"Dimension" -> 2}, ImageSize -> {100, 100}, PlotRange -> All, PlotTheme -> "Monochrome", VertexCoordinates -> {{0.7071067811865475, -0.7071067811865475}, {0.3743506488634663, -0.23570226039551573}, {0.7071067811865475, 0.7071067811865475}, {-0.7071067811865475, -0.7071067811865475}, {0.23570226039551578, -0.3743506488634662}, {0.18024290500833565, -0.09705387192756525}, {0.37435064886346636, 0.23570226039551584}, {0.23570226039551584, 0.3743506488634663}, {-0.7071067811865475, 0.7071067811865475}, {-0.37435064886346625, -0.23570226039551576}, {-0.23570226039551578, -0.37435064886346625}, {0.09705387192756534, -0.18024290500833556}, {0.06932419423397529, -0.06932419423397518}, {0.18024290500833567, 0.09705387192756536}, {0.09705387192756541, 0.18024290500833562}, {-0.2357022603955157, 0.37435064886346625}, {-0.37435064886346625, 0.23570226039551578}, {-0.18024290500833556, -0.0970538719275653}, {-0.0970538719275653, -0.18024290500833556}, {-0.06932419423397519, -0.06932419423397518}, {0.0693241942339753, 0.06932419423397525}, {-0.09705387192756523, 0.18024290500833556}, {-0.18024290500833554, 0.09705387192756532}, {-0.06932419423397516, 0.06932419423397523}}, VertexSize -> {{0.01, 0.01}}, VertexStyle -> {GrayLevel[0.6]}}]

Findcomm[G1]

True.

G2=Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, {Null, SparseArray[Automatic, {24, 24}, 0, {1, {{0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96}, {{2}, {3}, {4}, {5}, {1}, {5}, {6}, {7}, {1}, {4}, {7}, {8}, {1}, {3}, {9}, {10}, {1}, {2}, {10}, {11}, {2}, {11}, {12}, {13}, {2}, {3}, {13}, {14}, {3}, {9}, {14}, {15}, {4}, {8}, {10}, {16}, {4}, {5}, {9}, {17}, {5}, {6}, {17}, {18}, {6}, {18}, {19}, {20}, {6}, {7}, {20}, {21}, {7}, {8}, {21}, {22}, {8}, {16}, {22}, {23}, {9}, {15}, {17}, {19}, {10}, {11}, {16}, {18}, {11}, {12}, {17}, {19}, {12}, {16}, {18}, {23}, {12}, {13}, {23}, {24}, {13}, {14}, {22}, {24}, {14}, {15}, {21}, {24}, {15}, {19}, {20}, {24}, {20}, {21}, {22}, {23}}}, Pattern}]}, {FormatType -> TraditionalForm, GraphLayout -> {"Dimension" -> 2}, ImageSize -> {100, 100}, PlotRange -> All, PlotTheme -> "Monochrome", VertexCoordinates -> {{0.40394241242329176, -0.25945599573690153}, {0.7071067811865475, -0.7071067811865475}, {0.37907330901109937, 0.1313241832557828}, {0.2620631824699983, -0.0838523166788216}, {0.2675263770255219, -0.3781890683380199}, {-0.7071067811865475, -0.7071067811865475}, {0.7071067811865475, 0.7071067811865475}, {0.1431808599645599, 0.16149826425230693}, {0.1221576317492572, -0.03377681913464458}, {0.143079376696345, -0.17350063509952313}, {-0.18402306220409667, -0.37269286132910723}, {-0.3833586543598687, -0.187026754982597}, {-0.7071067811865475, 0.7071067811865475}, {0.13147477516274866, 0.4022338079342901}, {-0.05998227606486567, 0.14621188495379928}, {-0.05969289213387439, -0.03925258901254048}, {-0.07942968445939742, -0.19818433624660647}, {-0.21708216019596363, -0.2072912595452552}, {-0.2215172397604917, -0.07126108562271007}, {-0.38772843629647186, 0.2375521064241246}, {-0.22690672076222454, 0.42625458277242784}, {-0.09748181973788815, 0.3140756035258782}, {-0.22593525235226009, 0.1485262610495524}, {-0.2345130572872111, 0.28160213844299575}}, VertexSize -> {{0.01, 0.01}}, VertexStyle -> {GrayLevel[0.6]}}]

Findcomm[G2]
Flase